structured/0710.3804.structured.json

{"paper_meta":{"paper_id":"arxiv:0710.3804","title":"0710.3804","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0710.3804v2 [cs.CC] 28 Jan 2008\nNoname manuscript No.\n(will be inserted by the editor)\nThierry Mora · Lenka Zdeborov ́a\nRandom subcubes as a toy model for\nconstraint satisfaction problems\nReceived: date / Accepted: date\nAbstract We present an exactly solvable random-subcube model inspired\nby the structure of hard constraint satisfaction and optimization problems.\nOur model reproduces the structure of the solution space of the random k-\nsatisfiability and k-coloring problems, and undergoes the same phase tran-\nsitions as these problems. The comparison becomes quantitative in the\nlarge-k limit. Distance properties, as well the x-satisfiability threshold, are\nstudied. The model is also generalized to define a continuous energy land-\nscape useful for studying several aspects of glassy dynamics.\nKeywords Constraint satisfaction problems · Clustering of solutions ·\nExactly solvable models\n1 Introduction\nCombinatorial optimization and Constraint Satisfaction Problems (CSPs)\narise in a wide array of scientific branches, including statistical physics,\ninformation theory, inference and machine learning. These problems, which\ninvolve a large number of variables interacting through a large number of\nconstraints or cost terms, are in general very hard to solve, and in most\nT. Mora\nLewis Sigler Institute for Integrative Genomics, Princeton University, Princeton,\nNJ 08544, USA\nL. Zdeborov ́a\nUniversit ́e Paris-Sud, LPTMS, UMR8626, Bˆat. 100, Universit ́e Paris-Sud 91405\nOrsay cedex, France\nCNRS, LPTMS, UMR8626, Bˆat. 100, Universit ́e Paris-Sud 91405 Orsay cedex,\nFrance\n\n2\ncases no algorithm seems to be able to find a solution within a reasonable\ntime, as formalized by the P ̸= NP conjecture [1].\nIn order to circumvent this intrinsic difficulty and possibly to identify\n(or avoid) “hard” instances, random ensembles of optimization problems\nwere introduced and used as test beds for theories and algorithms. This\nline of research has considerably benefited from the methods and concepts\nof statistical mechanics [2,3,4,5]. In particular, a spectacular breakthrough\nwas made by the development of the survey propagation algorithm [5,6]\nwhich is able to solve large random instances of CSPs in the so-called\n“hard-SAT” region. The key to the success of the physics approach lies in\nthe understanding of the rugged energy landscape (reminiscent of glassy\nphases) exhibited by these problems, which survey propagation exploits\nand integrates into a sophisticated message-passing [7] scheme.\nThe structure and organisation of solutions has been analyzed in detail\nfor several CSPs, including the satisfiability problem (k-SAT) [8], the col-\norability of random graphs (k-COL) [9,8,10], and systems of linear Boolean\nequations (k-XORSAT) [11,12,13]. It was shown that as the density of con-\nstraints is increased the space of solutions undergoes several phase transi-\ntions. At low density of constraints the solution space is concentrated in\none big ergodic component, called “cluster” or “state”. For higher densi-\nties the systems undergoes a clustering transition, whereby the solution\nspace breaks into an exponential number of well-separated clusters (this\nseparation can be energetic or entropic). For even higher densities a second\ntransition occurs, which reduces these clusters to a finite number. Finally,\nall clusters disappear at the SAT-UNSAT threshold.\nSince the clustering phenomenon is one of the main building blocks\nunderlying the statistical physics approach1, substantial efforts have been\nmade to give a rigorous base to it [15,16]. Some mathematical results were\nalso obtained in the ergodic phase [17], and in the simple case of linear\nBoolean equations [18]. Remarkably, pure states play an central role in\nspin-glass theory, and they have been extensively studied in that context.\nHowever, the geometrical organization of glassy phases is not yet fully\nunderstood, and the classical picture of complex energy landscapes with\nmany “valleys” still lacks an appropriate representation.\nIn this paper we introduce an exactly solvable random-subcube model2\n(RSM), in the spirit of Derrida’s Random Energy Model (REM) [19]. This\nmodel is inspired by the structure of hard CSPs and optimization prob-\nlems, and reproduces most of their phenomenology. It can be also thought\nof as an attempt to construct a minimal setting that is able to reproduce\nthe structure of solutions in hard CSPs. Its purpose is mainly pedagogi-\n1 Technically, clustering is closely related to the one-step replica-symmetry\nbreaking ansatz used in the replica/cavity method [14,6].\n2 Independently of our work A. Montanari inspired by an idea of D. Achlioptas\nalso introduced this model and worked out some parts of our Sec. 2.\n\n3\ncal, and it offers an excellent testing playground for ideas and methods in\ncombinatorial optimization and glass physics, while being fully tractable.\nDespite its simplicity, the RSM undergoes the same phase transitions as\nthose observed in random CSPs such as k-SAT [6,8] or k-COL [8,10]. The\nconnection even becomes quantitative in the large-k limit of these problems.\nSo far only the zeroth order of this limit was intuitively known and related\nto Shannon’s random code model [20,21,22], in which clusters are uniformly\ndistributed singletons. One of our most notable results is that the RSM\nprovides the first-order approximation of this large-k limit, reproducing\nthe cluster size distribution and freezing properties of the original models.\nWe also generalize the RSM to deal with continuous energy landscapes\nresembling those observed in glassy systems and hard optimization prob-\nlems, and show how static and dynamical properties can be related explic-\nitly. This energetic RSM displays temperature chaos, undergoes a dynam-\nical transition, and has a Kauzmann temperature.\nThe paper is organized as follows: In Sec. 2 we define the model and\ndescribe its basic properties, as well as the connection with the random k-\nSAT and k-COL problems. We also analyze the behaviour of physics-guided\ndecimation schemes. In Sec. 3 we discuss the relation between its dynamical\nand geometrical properties. In Sec. 4 we extend the definition to energetic\nlandscapes, compute static and dynamical properties, and comment on\nsome ideas from the physics of glassy systems. Finally we present a general\ndiscussion of our results in Sec. 5.\n2 The random-subcube model\n2.1 Definition\nMost constraint satisfaction problems are defined by a set of constraints\non N variables σ = (σ1, . . . , σN) with a finite alphabet, e.g. {0, 1}. In\ncontrast, the random-subcube model is defined directly by its solution space\nS ⊂{0, 1}N; we define S as the union of ⌊2(1−α)N⌋random clusters (where\n⌊x⌋denotes the integer value of x). A random cluster A being defined as:\nA = {σ | ∀i ∈{1, . . . , N}, σi ∈πA\ni },\n(1)\nwhere πA is a random mapping:\nπA : {1, . . . , N} −→{{0}, {1}, {0, 1}}\n(2)\ni 7−→πA\ni\n(3)\nsuch that for each variable i, πA\ni\n= {0} with probability p/2, {1} with\nprobability p/2, and {0, 1} with probability 1−p. A cluster is thus a random\nsubcube of {0, 1}N. If πA\ni\n= {0} or {1}, variable i is said “frozen” in A;\n\n4\notherwise it is said “free” in A. One given configuration σ might belong to\nzero, one or several clusters. A solution belongs to at least one cluster.\nThe parameter α is analogous to the density of constraints in CSPs;\nclearly the SAT-UNSAT transition occurs at αs := 1, where clusters cease\nto exist. The parameter p gives the probability that a variable is frozen,\nand plays a role similar to the clause size k in k-SAT, or to the number\nof colors k in k-coloring, as we will see later. Note that in the special case\np = 1, the RSM is equivalent to the random code model with rate R = 1−α\n[21].\nFrozen variables and the structure of the solution space have been in-\ntroduced to mimic the situation observed in random CSPs. However there\nare important differences between the RSM and models like k-SAT. First,\nin real CSPs the clusters are not necessarily subcubes of {0, 1}N. We stress\nhere that when speaking about clusters in the RSM we have in mind the\nabove definition, whereas in the context of the CSPs the notion of cluster is\nmore general [5,8]. Further, in real CSPs the sets of frozen variables associ-\nated with clusters are correlated by the underlying graph, instead of being\ndistributed uniformly. Moreover, free variables in CSPs do not enjoy the\nsame freedom as in the RSM, as clusters usually do not fill up the whole\nsubcube allowed by the frozen variables. In fact, free variables can be corre-\nlated within each cluster in a highly nontrivial way, and these correlations\nmay even be so strong that they create clusters without the help of frozen\nvariables. Clusters without frozen variables are indeed very important, as\ndiscussed recently in [10,23].\n2.2 The basic structural phase transitions\nWe now describe the static properties of the RSM in the thermodynamic\nlimit N →∞(the two parameters α and p being fixed and independent\nof N). The internal entropy s of a cluster A is defined as\n1\nN log2 |A|, i.e.\nthe fraction of free variables in A. The probability P(s) that a cluster has\ninternal entropy s follows the binomial distribution\nP(s) =\n N\nsN\n \n(1 −p)sNp(1−s)N .\n(4)\nLet N(s) be number of clusters of entropy s. This number follows a binomial\nlaw of parameter P(s) with 2N(1−α) terms. Then the mean and the variance\nof N(s) read:\nEN(s) = 2N(1−α)P(s) ,\nVarN(s) = 2N(1−α)P(s)[1 −P(s)] .\n(5)\nBy Markov’s inequality:\nP [N(s) ≥1] ≤E [N(s)] ,\n(6)\n\n5\nand by Chebyshev’s inequality:\nP\n \nN(s)\nEN(s) −1\n > ε\n \n≤\nVarN(s)\n[EN(s)]2ε2 ≤\n1\n2N(1−α)ε2P(s)\nfor all ε > 0,\n(7)\nwe get, with high probability (w.h.p.: with probability going to 1 as N →\n∞):\nlim\nN→∞\n1\nN log2 N(s) =\n \nΣ(s) := 1 −α −D(s ∥1 −p) if Σ(s) ≥0,\n−∞\notherwise,\n(8)\nwhere D(x ∥y) := x log2\nx\ny +(1−x) log2\n1−x\n1−y is the binary Kullback-Leibler\ndivergence. Throughout the paper, the same Markov/Chebyshev argument\nwill apply every time we will have to deal with a number of clusters with\na specific property.\nWe now compute the total entropy stot =\n1\nN log2 |S|. First note a ran-\ndom configuration belongs on average to 2N(1−α)(1 −p\n2)N clusters. There-\nfore, if\nα < αd := log2 (2 −p),\n(9)\nthen with high probability the total entropy is stot = 1.\nNow assume α > αd. The total entropy is given by a saddle-point\nestimation:\nX\nA\n2s(A)N = [1 + o(1)]N\nZ\nΣ(s)≥0\nds 2N[Σ(s)+s],\n(10)\nwhence\nstot = max\ns\n[Σ(s) + s | Σ(s) ≥0].\n(11)\nWe denote by s∗= argmax[. . . ] the fraction of free variables in the clusters\nthat dominate the sum. Note that in this sum solutions belonging to several\nclusters have been counted too many times. This does not affect the validity\nof our estimation, since in every cluster the fraction of solutions belonging\nto more than one cluster is exponentially small as long as α > αd.\nDefine ̃s := 2(1 −p)/(2 −p) such that ∂sΣ( ̃s) = −1. The complexity of\nclusters with entropy ̃s reads:\nΣ( ̃s) =\np\n2 −p + log2(2 −p) −α.\n(12)\n ̃s maximizes Eq. (11) as long as Σ( ̃s) ≥0, that is if\nα ≤αc :=\np\n(2 −p) + log2 (2 −p).\n(13)\nThen the total entropy reads\nstot = 1 −α + log2 (2 −p)\nfor\nα ≤αc.\n(14)\n\n6\n-0.05\n0\n0.05\n0.1\n0.15\n0.2\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\ns\nslope −1\nsM\nstot\n-0.01\n0\n0.01\n0.02\n0.03\n0.04\n0.45\n0.5\n0.55\n0.6\n0.65\ns\nslope −1\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.8\n0.85\n0.9\n0.95\n1\n1\n0.8\n0.6\n0.4\n0.2\nΣ∗\nΣtot\nm\nstot = sM\nslope −m\nstot\nαc\nNon-Condensed\nCondensed\ns∗\nΣ∗\ns, Σ\nm\ns∗\ns∗= sM\nΣ(s)\nΣ(s)\nα\nFig. 1 (Color online) Top: graphical construction of the maximum of Σ(s) + s\nby a Legendre transformation. In the top left figure (α < αc), the line of slope\n−1 tangent to the complexity function gives the saddle-point s∗, as well as the\ntotal entropy stot by the intercept on the s axis. In the top right figure (α > αc),\nthe supporting line of slope −1 gets “stuck” at sM, where the derivative is −m >\n−1. Bottom: represented as a function of α: total entropy stot, total complexity\nΣtot = 1 −α, typical entropy s∗, complexity of dominating clusters Σ∗= Σ(s∗),\nand m = −∂sΣ(s∗). The condensation point αc marks the separation between\nthe two regimes illustrated above.\nFor α > αc, the maximum in (11) is realized by the largest possible cluster\nentropy sM, which is given by the largest root of Σ(s). Then stot = s∗=\nsM. In this phase the dominating clusters3 have size eNs∗+∆, where ∆=\nO(1) is asymptotically distributed according to a Poisson point process of\nrate e−m∆, i.e., for d∆≪∆the probability that there is at least one state\nof size between eNs∗+∆and eNs∗+∆+d∆is e−m∆d∆, where m = −∂sΣ(s∗).\nExtreme value study of this process leads to the Poisson-Dirichlet [24,25,26]\ndistribution of weights of clusters. In particular it follows that an arbitrary\nlarge fraction of the solutions can be covered by a finite number of clusters.\nSuch a phase is called condensed.\nIn summary, for a fixed value of the parameter p, and for increasing\nvalues of α, four different phases can be distinguished:\n(a) Liquid phase, α < αd: almost all configurations are solutions.\n3 The “dominating clusters” are the minimal set of clusters covering almost all\nsolutions.\n\n7\n(b) Clustered phase with many states, αd < α < αc: an exponential number\nof clusters is needed to cover almost all solutions.\n(c) Condensed clustered phase, αc < α < αs = 1: a finite number of the\nbiggest clusters cover almost all solutions.\n(d) Unsatisfiable phase, α > αs: no cluster, hence no solution, exists.\nThe very same series of phase transitions is observed in random k-coloring\nand random k-satisfiability, where α is the density of constraints [8,10].\nThe condensation transition at αc corresponds to the Kauzmann temper-\nature [27] in the theory of glasses; at αc the total entropy stot(α) has a\ndiscontinuity in its second derivative with respect to α, in analogy with the\ndiscontinuity of the specific heat at the Kauzmann temperature.\nKeeping in mind the similarity between the RSM and real CSPs, it can\nbe useful to mention some of the properties that are commonly discussed\nin the statistical physics analysis of these problems (for brevity we omit the\nproves of these statements). Among them is the probability distribution of\nmutual overlaps P(q), where q(σ, σ′) = PN\ni=1[2δ(σi, σ′\ni) −1]/N. Below the\ncondensation transition, α < αc, we have P(q) = δ(q), reflecting the fact\nthat random pairs of solutions are uncorrelated. For α > αc, the overlap\nfunction consists of intra-cluster and inter-cluster overlaps: P(q) = wδ[q −\n(1 −stot)] + (1 −w)δ(q), where w is the sum of squares of weights of all\nthe clusters, it is a non-self-averaging random variable, the distribution of\nwhich can be computed from the Poisson-Dirichlet process [26,28].\nAn equivalent way of characterizing the condensed phase is to consider\nthe k-point correlation function, with k ≥2:\nX\nx1,...,xk\n|P(σ1 = x1, σ2 = x2, . . . , σk = xk) −P(σ1 = x1) . . . P(σk = xk)|.\n(15)\nThis quantity decays to zero as N goes to infinity in the non-condensed\nphase, whereas it remains bounded away from zero in the condensed phase.\n2.3 The large k-limit of random k-SAT and k-COL\nOne of the most interesting properties of the random-energy model [19] is\nits equivalence with the large-p limit of the p-spin glass [28]. In the same\nspirit, although the justification is slightly different, the random-subcube\nmodel is found to be equivalent to random k-SAT, random k-COL, and\npresumably other constraint satisfaction problems in the limit k →∞,\nfor connectivities close to the satisfiability threshold. Let us detail this\nstatement.\nIt was already known that at zeroth order (when k →∞) random k-\nSAT and k-COL behave as a random-code model (random-subcube model\nwith p = 1), in which clusters are uniformly distributed singletons. Recent\n\n8\nlarge-k calculations of the cluster size distribution Σ(s) in k-SAT and k-\nCOL [8,10] allow a direct comparison with the RSM.\nThe control parameters of the RSM are rescaled as:\np = 1 −ε ,\nα = 1 + ε1 + γ\nln 2 ,\n(16)\nwith ε ≪1 and γ = Θ(1). The cluster size distribution in the RSM is then,\nat leading order:\nΣ(s) ln(2) = s\nh\n1 −ln s\nε\ni\n−ε(2 + γ) + o(ε).\n(17)\nwhile the condensation and satisfiability thresholds read, in terms of the\nrescaled variable γ:\nγc = −2 ln 2 ,\nγs = −1.\n(18)\nIn k-SAT and k-COL, identifying ε and γ in the following way:\nSAT :\nε =\n1\n2k+1 ,\nM\nN = 2k ln 2 −ln 2\n2\n+ γ\n2 ,\n(19)\nCOL :\nε = 1\n2k ,\nM\nN = k ln k −ln k\n2\n+ γ\n2 ,\n(20)\nwhere M is the number of constraints (edges in coloring), and N the number\nof variables, gives a perfect match4 for the complexity function (17), as\ncomputed in [8,10]. This illustrates the analogy between α and the density\nof constraints M/N, as well as between p and k.\nThis equivalence goes further than simply having identical cluster size\ndistributions. The cavity analysis of k-SAT shows that the fraction of free\nvariables in a cluster scales exactly as its internal entropy s. The entropy\nof clusters is thus maximal, from which we infer that clusters fill up the\nwhole subcube prescribed by their frozen variables, like in the RSM. The\nsame is true for k-COL (compare Eqs. (E14) and (E27) in [10]) with the\nsmall difference that for every unfrozen variable only two (out of k) colors\nare allowed.\nNote that this comparison is valid only in a finite vicinity of the satis-\nfiability threshold, for γ = Θk(1). In particular, it does not encompass the\nclustering transition, which for k-SAT (resp. k-COL) occurs for constraint\ndensities scaling as 2k ln 2/k (resp. k ln k/2).\n4 Note the difference in the logarithmic base between here and [8,10].\n\n9\n2.4 Decimation\nAn important contribution of statistical physics to the field of combinatorial\noptimization has been to exploit the information provided by message-\npassing algorithms to devise physics-guided decimation schemes.\nMessage-passing algorithms exchange information between units (vari-\nables and constraints) in order to obtain estimates of marginal probabilities\n(beliefs) or other related quantities (e.g. surveys, see below). Subsequently,\nthis information is used to find a solution. A usual way to do this, called\ndecimation, proceeds as follows: fix randomly the value of one5 variable ac-\ncording to its estimated belief, then re-run the message-passing algorithm\non the reduced system, and loop. A trivial statement is that a perfect\nestimate of all marginal probabilities would always cause the decimation\nprocedure to find a solution (if any).\nIn the RSM there is no underlying graph, therefore message-passing can-\nnot be defined. However, it is possible to study decimation schemes based\non exact marginal probability estimators. Although such procedures should\nreally be viewed as thought experiments, they can be used to gain some\ninsight on real algorithms. Here two idealized algorithms are considered:\n– Belief estimator: outputs the exact marginal probabilities μi(σi) =\nP\nσ\\i μ(σ), where μ(σ) = I(σ ∈S)/|S|.\n– Survey estimator: outputs “surveys”, i.e. marginal probabilities over the\nclusters: νi(πi) = P\nπ\\i ν(π), where ν(π) = P\nA I(π = πA)/⌊2N(1−α)⌋.\nIn real CSPs, belief and survey propagation arguably provide asymp-\ntotically accurate estimators, as long as the number of clusters dominating\nthe measure μ (or ν for survey propagation) scales exponentially with N,\nand as long as the one-step replica symmetry breaking description is the\ncorrect one [14,10]. However, belief-guided and survey-guided decimation\nschemes are difficult to analyze in real CSPs — see [29] for an empirical\nstudy on surveys and [30] for recent analytical study on beliefs in k-SAT.\nLet us study decimation in the RSM. As long as the phase is non-\ncondensed, the belief estimator always outputs μi(σi) ≈1/2 for all i in\nthe limit N →∞. Likewise, the survey estimator will output νi({0}) ≈\nνi({1}) ≈p/2, νi({0, 1}) ≈1 −p. In both cases, the decimation procedure\nis completely unbiased: it will fix a random variable i to 0 or 1, with proba-\nbility 1/2. This observation remains true in the subsequent decimation steps\nas long as the number of clusters dominating the reduced measure μ (or\nν) remains exponential. Within this assumption, after T = tN (0 ≤t ≤1)\ndecimation steps, T variables will be fixed randomly and independently.\nWe are then left with a restricted space of solutions compatible with these\nT fixed variables. The logarithm of the number of clusters of entropy sN\n5 In practice the number of variables fixed at each step can range from one to\na small fraction of the variables.\n\n10\nis then\nNΣt(s) = N\n \n1 −α + t log2\n \n1 −p\n2\n \n−(1 −t)D\n \ns\n1 −t||1 −p\n \n.\n(21)\nRescaling by the number of unfixed variables (1 −t)N: Σt = (1 −t) ̄Σt,\ns = (1 −t) ̄s, we obtain\n ̄Σt( ̄s) = 1 −α −tαd\n1 −t\n−D( ̄s||1 −p).\n(22)\nThe parameter ̄α(t) := (α −tαd)/(1 −t) now plays the same role as α\nin the analysis of the RSM. Consequently, the system undergoes the same\ncondensation and unsatisfiability transitions as t is increased. Assume for\nexample that αd < α < αc. Fixing a fraction tc := (αc −α)/(αc −αd)\nof the variables will cause the system to condense. The belief estimator\nwill then be dominated by a finite number of clusters, yielding instance-\ndependent biases on variables. An extensive number of variables become\nsuddenly near-frozen, i.e., μi(1) ≈0 or ≈1, and remain so for t > tc. At\nts := (1 −α)/(1 −αd), the total complexity goes to zero, and near-frozen\nvariables become truly frozen, i.e. μi(1) = 0 or 1, as all sub-dominant\nclusters disappear. By contrast, the survey estimator will output unbiased\nmarginal probabilities as long as the total number of clusters is exponential\nin N, that is if t < ts. At t = ts, decimation concentrates on a single\ncluster, causing a freezing avalanche in the surveys, i.e. for each variable\ni, νi({0, 1}) = 1 or νi({0}) = 1 or νi({1}) = 1. For both estimators, any\nslight error at ts will cause the failure of the decimation process.\nNote that in real CSPs, belief propagation is not expected to be correct\nin the condensed phase, beyond tc; in fact, it will not detect condensation,\nnor near-frozen variables. For a recent study of the belief-propagation-\nguided decimation in k-SAT see [30]. Remarkably, in real CSPs, survey-\nguided decimation usually simplifies the problem: after a certain number\nof decimation steps, it outputs νi({0, 1}) = 1 for all i, and the problem can\nthen be passed over to a simple local search algorithm [6,29]. Although the\nRSM, which is intrinsically clustered, is unable to capture this property, it\nsheds some light on why decimation may not work in some cases.\n3 Distance and ergodicity\nThe geometrical organization of solutions to CSPs is thought to play an\nimportant role in setting intrinsic limits to the performance of search al-\ngorithms. In particular, the clustering phenomenon, by which the solution\nspace is fragmented into many connected components far from each other,\nhas been proposed by physicists as a possible explanation for the failure of\nmost known algorithms [4,5,6]. Also, the role of frozen variables was re-\ncently discussed in [10]. Conversely, the success of survey propagation [6] is\n\n11\nusually explained by the fact that it explicitly incorporates the existence of\nclusters. The separability of clusters has been proved in the k-SAT problem,\nin compliance with the predictions of statistical physics [15]. Despite this\nevidence, the precise relation between geometry and algorithms still lacks a\nrigorous base. The random-subcube model offers an excellent opportunity\nto study these questions in a well controlled framework.\n3.1 The dynamical transition\nLet us first argue why the clustering transition at αd, defined by Eq. (9),\nactually corresponds to what is commonly refered to as the dynamical\n(ergodicity breaking) transition in real CSPs [31,32,33,8,12,10]. To this\nend we study a uniform unbiased random walk on the space of solutions:\nat each step, one is allowed to move from one solution to the other by\nflipping only one spin. We choose this dynamics for the sake of simplicity,\nbut most of the arguments below hold for more general dynamical rules,\nlike for example the flipping of a sub-extensive number of spins at each\nstep.\nWe have already pointed out that an arbitrary configuration belongs\nto 2(1−α)N(1 −p/2)N different clusters w.h.p. if α < αd, and to none if\nα > αd. Therefore below αd almost all configurations are solutions, and\nany reasonable dynamics will explore the entire phase space uniformly.\nOn the other hand, when α > αd, solutions become exponentially rare.\nLet A be a cluster of internal entropy s. What is the probability that after t\nsteps, a random walker ends up in another cluster B of internal entropy s′?\nLet a denote the proportion of variables that are free in A and frozen in B.\nThe probability distribution of a is given by the total number of partitions\nof {1, . . . , N} into four categories: frozen in A and B, only frozen in A, only\nfrozen in B, and frozen neither in A nor B. This probability reads:\nq(a) =\n1\n N\nNs\n N\nNs′\n \nN!\n(Na)![N(s −a)]![N(1 −s′ −a)]![N(s′ −s + a)]!.\n(23)\nIn order for the walker to reach B from A, it has to match perfectly the\nprescriptions (freezings) of B on these aN variables. If t = Θ(N d), the prob-\nability of this happening is ≈t2−aN. Additionally, variables that are frozen\nin both clusters must coincide, so that A and B have a non-empty intersec-\ntion. For a random choice of B, this happens with probability 2N(s′+a−1).\nConsequently, the probability that the walker wanders in any other\ncluster of entropy s′ after t steps is union-bounded by:\nφ(s →s′) ≤2NΣ(s′) X\na\nq(a) t 2−aN 2N(s′+a−1)\n(24)\n\n12\nThe maximum of\n1\nN log q(a) tends to zero when N →∞, so that:\nlim sup\nN→∞\n1\nN log φ(s →s′) ≤Σ(s′) + s′ −1\n(25)\nThis quantity remains negative for all s′, as long as α > αd. Therefore,\nin this regime, hopping from one cluster to the other is very unlikely in\na sub-exponential number of steps, even though clusters are not disjoint.\nIn this sense we say that ergodicity is broken. Remarkably, the space of\nsolutions becomes non-ergodic as soon as it becomes non-trivial, at αd.\nLet us stress the importance of the entropic barriers between clusters\nin our analysis. In the real CSPs and optimization problems the energetic\nbarriers are usually thought of as more important, and clusters are even\nsometimes described as separated (by an extensive distance). The impor-\ntance of entropic barriers and the possibility of non-extensively separated\nclusters should, however, not be neglected in the studies of richer models.\n3.2 x-satisfiability\nThe notion of x-satisfiability was first introduced as a tool to study the\ngeometrical structure of the solution space of CSPs [15]. An instance of CSP\nis said x-satisfiable if and only if it admits a pair of solutions separated by a\nHamming distance ∼xN. In other words, x-satisfiability gives the distance\nspectrum of the solution space. This spectrum is estimated using three\nquantities:\na) d1 = x1(α)N: the maximum distance between two solutions inside one\ncluster,\nb) d2 = x2(α)N: the minimum distance between two solutions from two\ndistinct clusters,\nc) d3 = x3(α)N: the maximum distance between any two solutions (pre-\nsumably from two different clusters).\nThe first of these quantities is estimated by noting that the maximum\ndistance between any two solutions in a given cluster, i.e. its diameter,\nequals its entropy s. Therefore the maximum diameter/entropy x1 is given\nw.h.p. by the largest internal entropy sM, i.e. the largest root of Σ(s) =\n1 −α −D(s ∥1 −p).\nNow take two clusters A and B at random, and consider the probability\nthat their distance be xN. This distance is given by the number of variables\nwhich are frozen in both clusters, but in a contradictory way, such that\nπA(i) ̸= πB(i). This happens independently with probability p2/2 for each\nvariable, so that the number of such variables follows a binomial law of\nparameter p2/2. Therefore, the number N(x) of pairs of clusters at distance\n\n13\nαs(x)\nx1\nx2\nαgap\nαsep\nαc\nx\nα\n1\n0.9\n0.8\n0.7\n0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0\n1\n0.9\n0.8\n0.7\n0.6\nFig. 2 (Color online) The x-satisfiability threshold is constructed from the three\ndistances x1(α), x2(α) and x3(α). Below a threshold αgap, distance spectra fail to\ndetect the fragmentation of the solution space (there is no more “gap” between\nintra and inter-cluster distances). Below another threshold αsep, clusters cease\nto be all well separated, although ergodicity is still broken. The condensation\nthreshold αc is shown for information. The dynamical threshold αd lies outside\nthe picture, and its value is ≈0.070. In this figure p = 0.95.\nxN coincides w.h.p. with its mean value:\nE[N(x)] = 22(1−α)N\n N\nNx\n \n1 −p2\n2\n (1−x)N p2\n2\n xN\n≍2Ns2(x)\n(26)\nif s2(x) := 2(1 −α) −D(x ∥p2/2) > 0, and N(x) = 0 w.h.p. if s2(x) < 0.\nConsequently the smallest possible distance between any two clusters is\ngiven by x2N, where x2(α) is the smallest root of s2(x). A similar argument\ngives the largest distance between any two solutions from two different\nclusters: x3(α) = 1 −x2(α).\nTo sum up, we find that a random instance is x-satisfiable w.h.p. if\nα < αs(x), and is x-unsatisfiable w.h.p. if α > αs(x), with:\nαs(x) =\n \n \n \n \n \n \n \n1\nif x ∈[0, 1 −p] ∪[p2/2, 1 −p2/2]\n1 −D(x ∥1 −p)\nif x ∈[1 −p, x0]\n1 −1\n2D(x ∥p2/2)\nif x ∈[x0, p2/2]\n1 −1\n2D(1 −x ∥p2/2) if x ∈[1 −p2/2, 1]\n(27)\nwhere x0 is solution to D(x ∥p2/2) = 2D(x ∥1 −p).\n\n14\nFig. 2 shows how αs(x) can be constructed from the three distances x1,\nx2, x3. We put αsep := 1 + (1/2) log(1 −p2/2) > αd, the threshold below\nwhich some pairs of clusters have a non-empty intersection. An interesting\nobservation is that ergodicity can still be broken even below this threshold.\nWe can also define the αgap := αs(x0) > αsep, below which distances\nfrom the same cluster and distances from distinct clusters overlap. This\nthreshold sets the limit below which the notion of x-satisfiability fails to\ndetect clustering. The random-subcube model allows us to make a clear\nand intelligible distinction between the three thresholds αd < αsep < αgap.\nWe expect this distinction to hold in most CSPs6.\n4 Random energy landscape\n4.1 Definition\nThe random-subcube model can be enriched by adding the notion of energy\nto the definition of states. The motivation for doing this is to mimic the\noptimization version of CSPs (where energy is defined as the number of un-\nsatisfied constraints), but it can also be used to reproduce some properties\nof glassy systems.\nFor each energy level E0, we define N(E0) = ⌊2NΣ(E0/N)⌋valleys of\nenergy E0, where Σ(e0) is an increasing complexity function. This function\ncan be arbitrary, but for simplicity we will restrict our examples to the\nform7 Σ(e0) = a + be0 −ce0 ln(e0), where a > 0 corresponds to a SAT\nphase, and a < 0 to an UNSAT phase.\nEach valley is defined as a subcube V , chosen at random in the same\nway as clusters in the previous sections, cf. (1). In the following, the freezing\nprobability p will be fixed for all energies, but one could easily generalize\nthe model by making it energy-dependent: p = p(e0).\nThe number N(E0, S0) of valleys of energy E0 = e0N and entropy\nS0 = s0N is w.h.p.:\nN(E0, S0) ≍2NΣ(e0,s0) if Σ(e0, s0) := Σ(e0) −D(s0 ∥1 −p) ≥0\n= 0\notherwise.\n(28)\nGiven a configuration σ, we define its energy as a trade-offbetween the\nenergy of surrounding valleys and their distance. Let us denote the energy\nof a valley V by E0(V ). Then the energy of σ is:\nE(σ) := min\nV\n[E0(V ) + d(σ, V )]\n(29)\n6 With the notable exception of k-XORSAT, where αd = αsep. Incidentally in\nk-XORSAT we also have αc = αs.\n7 This form corresponds to the function used to fit data from the cavity method\nin the k-SAT problem [6].\n\n15\nwhere d(σ, V ) is the distance between σ and the nearest element of V .\nBy definition, we say that σ belongs to the basin of attraction of V if\nV minimizes the sum. Observe that with this definition, it may happen\nthat some valleys are not represented at all in the energy landscape. In the\nfollowing, the term “state-bottom energy” shall refer to the energy E0 of\nthe valley minimizing the sum, and the term “state-bottom entropy” to the\nentropy S0 of that valley.\n4.2 Static description of the energy landscape\nWhat is the energy of an arbitrary configuration σ? Let us start with\nthe typical case: for each e0 = E0/N, we compute the distance to the\nnearest valley with energy E0. Standard arguments show that the number\nof valleys of state-bottom energy e0N at distance d = ωN is governed by\nthe exponent:\nΣ(e0) −D(ω ∥p/2).\nThen, the minimum distance is given, in the N\n→\n∞limit, by\nδ[Σ(e0), p/2]N, where δ(x, y) is solution to: x = D(δ ∥y). Then, the typi-\ncal energy is obtained as the best compromise between state-bottom energy\nand distance:\ne∗= E/N = min\ne0 {e0 + δ[Σ(e0), p/2]} ,\n(30)\nThe argmin gives the typical state-bottom energy e∗\n0 of a random σ.\nAs we just saw, most configurations have roughly the same energy, and\nbelong to valleys with the same state-bottom energy. At finite temperature\nhowever, thermodynamics will be dominated by configurations of lower\nenergy than e∗N. We thus need to estimate the entropy function (governing\nthe number of configurations of given energy), the Legendre transform of\nwhich shall give us the free energy. Given a valley V of energy E0 = e0N\nand entropy S0 = s0N, the number of configurations of energy E belonging\nto this valley is:\nNV (E) = 2S0\n E −E0\nN −S0\n \n,\n(31)\n≍2NsV (e|e0,s0),\nwith\nsV (e|e0, s0) := s0 + (1 −s0)H\n e −e0\n1 −s0\n \n,\nwhere H(x) = −x log2 x−(1−x) log2 (1 −x) is the entropy function. Sum-\nming up over all valleys, the total number of configurations with energy\nE = eN is:\nN(E) =\nX\nS0,E0\n2S0\n E −E0\nN −S0\n \n2NΣ(E0/N,S0/N) ≍2Ns(e),\n(32)\nwith\ns(e) =\nmax\ne0,s0\nΣ(e0,s0)≥0\n[sV (e|e0, s0) + Σ(e0, s0)] .\n(33)\n\n16\nHere we have implicitly assumed that all elements in the sphere of radius\nE−E0 and center V are in the basin of attraction of V , as long as E < e∗N.\nThis is not true in general, as some configurations may in fact belong to\na more favorable basin, and may thus have lower energies. However, such\nconfigurations remain exponentially rare in comparison to the total weight\nof the sphere. Therefore the previous estimate holds8.\nCanceling the derivative w.r.t. s0 in (33) yields the saddle for s0:\n ̃s0 = (1 −p)(1 −e + e0)\n1 −p/2\n(34)\nProvided that the maximum is reached in a region where Σ(e0, ̃s0) > 0, we\nget:\ns(e) = max\ne0 [1 −D(e −e0 ∥p/2) + Σ(e0)]\nfor\nec < e < e∗.\n(35)\nThis will be valid from e = e∗(for which we find s(e∗) = 1 as expected)\ndown to a certain condensation energy ec. Below that energy (e ≤ec), the\nphase is condensed: Σ(e0, ̃s0) < 0, and the maximum in (33) is reached on\nthe border on the definition domain, where Σ(e0, s0) = 0. If we denote by\nsM(e0) the biggest valley of energy e0 (i.e. the largest of root of Σ(e0, s0) =\n0), we get:\ns(e) = max\ne0 {sV [e|e0, sM(e0)]}\nfor\ne < ec.\n(36)\nSo far we have worked in the microcanonical ensemble, but the same ar-\nguments hold in the canonical ensemble. In particular the condensation\ntemperature is Tc = (∂s/∂e|e=ec)−1. The number of dominating states in\nthe condensed phase follows again a Poisson-Dirichlet process [25,26] with\nparameter m, where m/T is the slope of the curve ΣT (f) at its smallest\nroot. The function ΣT (f) is the canonical counterpart of Σ(e0, s0), and\nreads:\nΣT (f) =\nmax\ne0,s0:fV (T |e0,s0)=f Σ(e0, s0) ,\n(37)\nwhere fV (T |e0, s0) is the single-state free energy, obtained as the Legendre\ntransform of sV (e|e0, s0) in (31).\nIn both low-temperature phases (condensed and non-condensed), equi-\nlibrium is reached for different values of (e0, s0) as e varies. Said differently,\nthe states dominating the microcanonical measure at e and at e + δe are\ncompletely distinct. In the canonical language, we say that the system ex-\nhibits temperature chaos [34,35]: slightly changing the temperature from T\nto T + δT dramatically modifies the free-energy landscape, reshuffling the\nordering of states. Consequently, correlations are nonexistent between T\n8 This argument is similar to the one used in Eq. (10), where some solutions\nwere counted several times, but with no consequence at the exponential scale.\n\n17\n 0.8\n 0.82\n 0.84\n 0.86\n 0.88\n 0.9\n 0.92\n 0.94\n 0.96\n 0.98\n 1\n 0.03\n 0.04\n 0.05\n 0.06\n 0.07\n 0.08\nV3\nV2\nV1\nsV2(e)\nsV3(e)\nslope 1/T\nsV1(e)\nsM(e0)\ns\ne\nFig. 3 (Color online) Illustration of the temperature chaos by construction of the\nmicrocanonical entropy in the condensed phase, cf. Eq. (36), and level crossing.\nWe have represented three entropy curves sV (e) corresponding to three extremal\nstates V1, V2 and V3, the envelope of all these curves is the microcanonical entropy.\nThese states are maximally atypical; they realize a balance between low state-\nbottom energies and a high state-bottom entropies, which are related by s0 =\nsM(e0) (thick curve). As the temperature (or energy) is decreased, the curves\nsV (e) cross each other, and the system is dominated by states of lower state-\nbottom energies and entropies. These data were obtained for number of valleys\nΣ(e0) = −0.05 −0.5e0 ln e0 and p = 0.2.\nand T +δT . This phenomenon of free-energy crossings is illustrated (in the\ncondensed phase) by Fig. 3, where the maximum of Eq. (36) is constructed\ngeometrically.\n4.3 Relation with dynamics\nWe now undertake to describe the dynamical properties of this energetic\nlandscape. To that end we shall make use of the static picture, which we\nknow precisely from bottom-up construction.\nOur reasoning proceeds in two steps. First, we study the behavior of\na single spin-flip Monte-Carlo dynamics with detailed balance evolving in\na single valley of state-bottom energy e∗\n0, and state-bottom entropy s∗\n0. In\na second step, we argue that the same dynamics run on the full rugged\nenergy landscape is entirely governed by this single-state-like behavior.\nThe thermodynamics of a single typical state is given by:\ns(e|e∗\n0, s∗\n0) = s∗\n0 + (1 −s∗\n0)H\n e −e∗\n0\n1 −s∗\n0\n \n(38)\n\n18\nwhere s∗\n0 = (1−p)(1−e∗+e∗\n0)/(1−p/2) is the typical state-bottom entropy\nof a random configuration, computed from Eq. (34). Supposing now that the\nenergy landscape is made of this state only, we can easily convince ourselves\nthat the dynamics is ergodic: the energy landscape is convex, and there are\nneither energetic nor entropic barriers. Both the quenched and annealed9\ndynamics in this simple landscape is then given by the thermodynamics.\nWe know from the previous paragraphs that states extend up to energy\ne∗, where a “crest” connects the different valleys. This crest is actually more\nof a plateau, as it embeds almost all configurations: s(e∗) = 1. Therefore,\nour dynamics will remain there as long as the temperature does not al-\nlow configurations of lower energy. This happens at Td := (∂s/∂e|e=e∗)−1,\nwhere exploring valleys starts to be more favorable.\nBelow Td, the system will find itself trapped in one state, since barri-\ners between valleys are extensive. This randomly picked state has typical\nproperties: in particular, its bottom-energy is e∗\n0 and its bottom-entropy is\ns∗\n0.\nWe argue that below Td everything happens as if this trapping state was\nput in isolation, as we have described above. The justification comes from\nthe fact that a valley does not “see” its neighbors as long as e < e∗: even\nthough many configurations of energy e < e∗in the isolated state actually\nbelong other valleys in the full landscape, their proportion is exponentially\nsmall. In other words, although there may be some directions for which the\nenergy barrier is lower than e∗−e, these directions are beaten entropically.\nWith this reasoning, both the quenched and annealed dynamics are\ndescribed by the microcanonical entropy of a single typical state: sdyn(e) =\ns(e|e∗\n0, s∗\n0). We note in passing:\nT −1\nd\n= ∂s\n∂e\n \ne=e∗= ∂sdyn\n∂e\n \ne=e∗,\n(39)\nas partial derivatives w.r.t. e0 and s inside (33) cancel at the maximum.\nThe dynamical temperature Td is thus well defined.\nNote that this analysis is exact: the fact that it purely relies on static\narguments should be attributed to the simplicity of the model. In richer\nmean-field models this kind of arguments might also apply, but will most\nprobably not give the full picture.\n4.4 Glassy behaviour\nThe analysis of the energy landscape in the RSM is summarized in the\nenergy-temperature phase diagram of Fig. 4. As anticipated, the behaviour\n9 By quench, resp. annealing, we mean a fast, resp. slow, change of temperature\nfrom T1 > Td to T2 < Td and some time (a finite number of Monte Carlo sweeps)\nspent at T2 afterwards.\n\n19\n0\n0.5\n1\nm\n0\n0.2\n0.4\n0.6\n0.8\n1\n1.2\n1.4\nT\n0\n0.2\n0.4\n0.6\n0.8\n1\n1.2\n1.4\nT\n0\n0.05\n0.1\n0.15\n0.2\n0.25\nenergy\ne∗\nec\negs\ne∗\n0\nTd\nTc\ncondensed glass\nglass\nliquid\ndynamical\nFig. 4 (Color online) Energy as a function of temperature. At temperature\nT > Td, the system is in a liquid state: the dynamics is exploring ergodically\nall configurations of energy e∗. Below the dynamical temperature ergodicity is\nbroken. The upper “dynamical” curve shows the result of a quench/annealing\nin temperature, whereby the systems remains stuck in a typical state of bot-\ntom energy e∗\n0. For Tc < T < Td, equilibrium thermodynamics is dominated\nby an exponential number of states (curve “glass”). Below the condensation\n(Kauzmann) temperature thermodynamics is dominated by a finite number of\nstates (curve “condensed glass”). This number is given by a Poisson-Dirichlet\nprocess of parameter m (plotted in the upper part of the diagram). The dashed\nline shows the result of a quench/annealing starting from an equilibrium state\nat temperature Tc < T < Td. The bottom line shows the bottom energies\nof the thermodynamically dominating states. These curves were obtained for\nΣ(e0) = −0.05 −0.5e0 ln e0 and p = 0.6.\nof the RSM resembles the one observed in glasses and spin-glasses. The\ntwo distinct glassy transitions (dynamical and condensation), as well as\nthe phenomenon whereby the physical dynamics gets stuck in metastable\nstates, have been described for example in the p-spin glass [28], the spherical\np-spin glass [36], the Potts glass [37] and the lattice glass [38]. Several\nrelated examples of energy-temperature diagrams were derived recently in\n[39].\nIn the aforementioned mean-field models, the static behaviour is bet-\nter understood than the dynamics. Static properties are usually analyzed\nby the replica/cavity method, with the help of Parisi’s replica symmetry\nbreaking scheme. A satisfactory analytic treatment of the dynamics exists\n\n20\nonly for the spherical p-spin glass [36], in which all states have the same\nentropy. A remarkable step towards connecting the static picture and dy-\nnamical behaviour in a rather general framework was done in [31,32,33].\nThe RSM could provide a tractable playground for studying several aspects\nof glassy dynamics, e.g. aging and rejuvenation [40].\nWe would like to emphasize the freedom we can enjoy in the definition\nof the energy landscape. First, arbitrary numbers and sizes of valleys at\neach energy e0, Σ(e0) and p(e0), can be considered. Second, the definition\nof the configurational energy E(σ) in Eq. (29) could be generalized to an\narbitrary function of all valleys V and σ; E(σ) = F({V }, σ). By tun-\ning these parameters, one could hope to reproduce the dynamics of more\ncomplex models on a quantitative level.\n5 Conclusions\nThe random-subcube model is a simple exactly solvable model capturing\nseveral interesting properties of random constraint satisfaction problems.\nRather than an attempt to construct a new realistic model for practical\ninstances of constraint satisfaction problems, it allows us to identify which\nproperties of random CSPs can be reproduced by a simple probabilistic\nstructure, and conversely, which of these properties may be intrinsically\nnon-trivial. Examples of reproducible properties include condensation, non-\nmonotony of the x-satisfiability threshold, temperature chaos and dynami-\ncal freezing in metastable states. From this point of view, the RSM stands\njust next to the random-energy model [19], the random-code model [20,21,\n22] or the random-energy random-entropy model [35].\nSince the relation between the RSM and the large-k limit of random\nk-SAT and k-COL is based on non-rigorous results from [8,10], it would\nbe interesting to establish this equivalence rigorously. Further, the RSM\nshould be helpful for understanding some properties which are too difficult\nto study in more realistic models, such as finite-size corrections or some\naspects of glassy dynamics.\nFinally, our work addresses the broad question of producing, infer-\nring, and representing complex and rugged structures of the hypercube.\nAlthough we retained the simplest choice of subcubes for clusters, more so-\nphisticated alternatives could be explored and used to reproduce detailed\ngeometrical features of solution spaces in CSPs.\n6 Acknowledgment\nWe would like to thank Florent Krz ֒aka la and Marc M ́ezard for fruitful\ndiscussions about this work, and Guilhem Semerjian and Jiˇr ́ı ˇCern ́y for\ncritical reading of the manuscript. This work has been partially supported\nby EVERGROW (EU consortium FP6 IST).\n\n21\nReferences\n1. C. H. Papadimitriou. Computational complexity. Addison-Wesley, 1994.\n2. S. Kirkpatrick and B. Selman. 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Probab., 25:855–900, 1997.\n27. W. Kauzmann. The nature of the glassy state and the behavior of liquids at\nlow temperatures. Chem. Rev., 43:219, 1948.\n28. D.J. Gross and M. M ́ezard. The simplest spin glass. Nucl. Phys. B, 240:431,\n1984.\n29. G. Parisi.\nSome remarks on the survey decimation algorithm for k-\nsatisfiability. arXiv:cs/0301015, 2003.\n30. A. Montanari, F. Ricci-Tersenghi, and G. Semerjian.\nSolving con-\nstraint satisfaction problems through belief propagation-guided decimation.\narXiv:0709.1667v1 [cs.AI], 2007.\n31. A. Montanari and G. Semerjian. From large scale rearrangements to mode\ncoupling phenomenology. Phys. Rev. Lett., 94:247201, 2005.\n32. A. Montanari and G. Semerjian. On the dynamics of the glass transition on\nbethe lattices. J. Stat. Phys., 124:103–189, 2006.\n33. A. Montanari and G. Semerjian. Rigorous inequalities between length and\ntime scales in glassy systems. J. Stat. Phys., 125:23, 2006.\n34. A. J. Bray and M. A. Moore. Chaotic nature of the spin-glass phase. Phys.\nRev. Lett., 58:57–60, 1987.\n35. F. Krzakala and O. C. Martin. Chaotic temperature dependence in a model\nof spin glasses. Eur. Phys. J. B., 28:199–208, 2002.\n36. L. F. Cugliandolo and J. Kurchan. Analytical solution of the off-equilibrium\ndynamics of a long-range spin glass model. Phys. Rev. Lett., 71:173, 1993.\n37. D. J. Gross, I. Kanter, and H. Sompolinsky. Mean-field theory of the potts\nglass. Phys. Rev. Lett., 55(3):304–307, Jul 1985.\n38. G. Biroli and M. M ́ezard. Lattice glass models. Phys. Rev. Lett., 88:025501,\n2002.\n39. F. Krzakala and J. Kurchan. A landscape analysis of constraint satisfaction\nproblems. Phys. Rev. B, 76:021122, 2007.\n40. J.-L. Barrat, M.V. Feigelman, J. Kurchan, and J. Dalibard. Slow Relaxations\nand Nonequilibrium Dynamics in Condensed Matter Les Houches Session\nLXXVII, 1-26 July, 2002. Springer, Berlin, 2003.","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0710.3804v2 [cs.CC] 28 Jan 2008\nNoname manuscript No.\n(will be inserted by the editor)\nThierry Mora · Lenka Zdeborov ́a\nRandom subcubes as a toy model for\nconstraint satisfaction problems\nReceived: date / Accepted: date\nAbstract We present an exactly solvable random-subcube model inspired\nby the structure of hard constraint satisfaction and optimization problems.\nOur model reproduces the structure of the solution space of the random k-\nsatisfiability and k-coloring problems, and undergoes the same phase tran-\nsitions as these problems. The comparison becomes quantitative in the\nlarge-k limit. Distance properties, as well the x-satisfiability threshold, are\nstudied. The model is also generalized to define a continuous energy land-\nscape useful for studying several aspects of glassy dynamics.\nKeywords Constraint satisfaction problems · Clustering of solutions ·\nExactly solvable models\n1 Introduction\nCombinatorial optimization and Constraint Satisfaction Problems (CSPs)\narise in a wide array of scientific branches, including statistical physics,\ninformation theory, inference and machine learning. These problems, which\ninvolve a large number of variables interacting through a large number of\nconstraints or cost terms, are in general very hard to solve, and in most\nT. Mora\nLewis Sigler Institute for Integrative Genomics, Princeton University, Princeton,\nNJ 08544, USA\nL. Zdeborov ́a\nUniversit ́e Paris-Sud, LPTMS, UMR8626, Bˆat. 100, Universit ́e Paris-Sud 91405\nOrsay cedex, France\nCNRS, LPTMS, UMR8626, Bˆat. 100, Universit ́e Paris-Sud 91405 Orsay cedex,\nFrance"},{"paragraph_id":"p2","order":2,"text":"2\ncases no algorithm seems to be able to find a solution within a reasonable\ntime, as formalized by the P ̸= NP conjecture [1].\nIn order to circumvent this intrinsic difficulty and possibly to identify\n(or avoid) “hard” instances, random ensembles of optimization problems\nwere introduced and used as test beds for theories and algorithms. This\nline of research has considerably benefited from the methods and concepts\nof statistical mechanics [2,3,4,5]. In particular, a spectacular breakthrough\nwas made by the development of the survey propagation algorithm [5,6]\nwhich is able to solve large random instances of CSPs in the so-called\n“hard-SAT” region. The key to the success of the physics approach lies in\nthe understanding of the rugged energy landscape (reminiscent of glassy\nphases) exhibited by these problems, which survey propagation exploits\nand integrates into a sophisticated message-passing [7] scheme.\nThe structure and organisation of solutions has been analyzed in detail\nfor several CSPs, including the satisfiability problem (k-SAT) [8], the col-\norability of random graphs (k-COL) [9,8,10], and systems of linear Boolean\nequations (k-XORSAT) [11,12,13]. It was shown that as the density of con-\nstraints is increased the space of solutions undergoes several phase transi-\ntions. At low density of constraints the solution space is concentrated in\none big ergodic component, called “cluster” or “state”. For higher densi-\nties the systems undergoes a clustering transition, whereby the solution\nspace breaks into an exponential number of well-separated clusters (this\nseparation can be energetic or entropic). For even higher densities a second\ntransition occurs, which reduces these clusters to a finite number. Finally,\nall clusters disappear at the SAT-UNSAT threshold.\nSince the clustering phenomenon is one of the main building blocks\nunderlying the statistical physics approach1, substantial efforts have been\nmade to give a rigorous base to it [15,16]. Some mathematical results were\nalso obtained in the ergodic phase [17], and in the simple case of linear\nBoolean equations [18]. Remarkably, pure states play an central role in\nspin-glass theory, and they have been extensively studied in that context.\nHowever, the geometrical organization of glassy phases is not yet fully\nunderstood, and the classical picture of complex energy landscapes with\nmany “valleys” still lacks an appropriate representation.\nIn this paper we introduce an exactly solvable random-subcube model2\n(RSM), in the spirit of Derrida’s Random Energy Model (REM) [19]. This\nmodel is inspired by the structure of hard CSPs and optimization prob-\nlems, and reproduces most of their phenomenology. It can be also thought\nof as an attempt to construct a minimal setting that is able to reproduce\nthe structure of solutions in hard CSPs. Its purpose is mainly pedagogi-\n1 Technically, clustering is closely related to the one-step replica-symmetry\nbreaking ansatz used in the replica/cavity method [14,6].\n2 Independently of our work A. Montanari inspired by an idea of D. Achlioptas\nalso introduced this model and worked out some parts of our Sec. 2."},{"paragraph_id":"p3","order":3,"text":"3\ncal, and it offers an excellent testing playground for ideas and methods in\ncombinatorial optimization and glass physics, while being fully tractable.\nDespite its simplicity, the RSM undergoes the same phase transitions as\nthose observed in random CSPs such as k-SAT [6,8] or k-COL [8,10]. The\nconnection even becomes quantitative in the large-k limit of these problems.\nSo far only the zeroth order of this limit was intuitively known and related\nto Shannon’s random code model [20,21,22], in which clusters are uniformly\ndistributed singletons. One of our most notable results is that the RSM\nprovides the first-order approximation of this large-k limit, reproducing\nthe cluster size distribution and freezing properties of the original models.\nWe also generalize the RSM to deal with continuous energy landscapes\nresembling those observed in glassy systems and hard optimization prob-\nlems, and show how static and dynamical properties can be related explic-\nitly. This energetic RSM displays temperature chaos, undergoes a dynam-\nical transition, and has a Kauzmann temperature.\nThe paper is organized as follows: In Sec. 2 we define the model and\ndescribe its basic properties, as well as the connection with the random k-\nSAT and k-COL problems. We also analyze the behaviour of physics-guided\ndecimation schemes. In Sec. 3 we discuss the relation between its dynamical\nand geometrical properties. In Sec. 4 we extend the definition to energetic\nlandscapes, compute static and dynamical properties, and comment on\nsome ideas from the physics of glassy systems. Finally we present a general\ndiscussion of our results in Sec. 5.\n2 The random-subcube model\n2.1 Definition\nMost constraint satisfaction problems are defined by a set of constraints\non N variables σ = (σ1, . . . , σN) with a finite alphabet, e.g. {0, 1}. In\ncontrast, the random-subcube model is defined directly by its solution space\nS ⊂{0, 1}N; we define S as the union of ⌊2(1−α)N⌋random clusters (where\n⌊x⌋denotes the integer value of x). A random cluster A being defined as:\nA = {σ | ∀i ∈{1, . . . , N}, σi ∈πA\ni },\n(1)\nwhere πA is a random mapping:\nπA : {1, . . . , N} −→{{0}, {1}, {0, 1}}\n(2)\ni 7−→πA\ni\n(3)\nsuch that for each variable i, πA\ni\n= {0} with probability p/2, {1} with\nprobability p/2, and {0, 1} with probability 1−p. A cluster is thus a random\nsubcube of {0, 1}N. If πA\ni\n= {0} or {1}, variable i is said “frozen” in A;"},{"paragraph_id":"p4","order":4,"text":"4\notherwise it is said “free” in A. One given configuration σ might belong to\nzero, one or several clusters. A solution belongs to at least one cluster.\nThe parameter α is analogous to the density of constraints in CSPs;\nclearly the SAT-UNSAT transition occurs at αs := 1, where clusters cease\nto exist. The parameter p gives the probability that a variable is frozen,\nand plays a role similar to the clause size k in k-SAT, or to the number\nof colors k in k-coloring, as we will see later. Note that in the special case\np = 1, the RSM is equivalent to the random code model with rate R = 1−α\n[21].\nFrozen variables and the structure of the solution space have been in-\ntroduced to mimic the situation observed in random CSPs. However there\nare important differences between the RSM and models like k-SAT. First,\nin real CSPs the clusters are not necessarily subcubes of {0, 1}N. We stress\nhere that when speaking about clusters in the RSM we have in mind the\nabove definition, whereas in the context of the CSPs the notion of cluster is\nmore general [5,8]. Further, in real CSPs the sets of frozen variables associ-\nated with clusters are correlated by the underlying graph, instead of being\ndistributed uniformly. Moreover, free variables in CSPs do not enjoy the\nsame freedom as in the RSM, as clusters usually do not fill up the whole\nsubcube allowed by the frozen variables. In fact, free variables can be corre-\nlated within each cluster in a highly nontrivial way, and these correlations\nmay even be so strong that they create clusters without the help of frozen\nvariables. Clusters without frozen variables are indeed very important, as\ndiscussed recently in [10,23].\n2.2 The basic structural phase transitions\nWe now describe the static properties of the RSM in the thermodynamic\nlimit N →∞(the two parameters α and p being fixed and independent\nof N). The internal entropy s of a cluster A is defined as\n1\nN log2 |A|, i.e.\nthe fraction of free variables in A. The probability P(s) that a cluster has\ninternal entropy s follows the binomial distribution\nP(s) =\n N\nsN"},{"paragraph_id":"p5","order":5,"text":"(1 −p)sNp(1−s)N .\n(4)\nLet N(s) be number of clusters of entropy s. This number follows a binomial\nlaw of parameter P(s) with 2N(1−α) terms. Then the mean and the variance\nof N(s) read:\nEN(s) = 2N(1−α)P(s) ,\nVarN(s) = 2N(1−α)P(s)[1 −P(s)] .\n(5)\nBy Markov’s inequality:\nP [N(s) ≥1] ≤E [N(s)] ,\n(6)"},{"paragraph_id":"p6","order":6,"text":"5\nand by Chebyshev’s inequality:\nP"},{"paragraph_id":"p7","order":7,"text":"N(s)\nEN(s) −1\n > ε"},{"paragraph_id":"p8","order":8,"text":"≤\nVarN(s)\n[EN(s)]2ε2 ≤\n1\n2N(1−α)ε2P(s)\nfor all ε > 0,\n(7)\nwe get, with high probability (w.h.p.: with probability going to 1 as N →\n∞):\nlim\nN→∞\n1\nN log2 N(s) ="},{"paragraph_id":"p9","order":9,"text":"Σ(s) := 1 −α −D(s ∥1 −p) if Σ(s) ≥0,\n−∞\notherwise,\n(8)\nwhere D(x ∥y) := x log2\nx\ny +(1−x) log2\n1−x\n1−y is the binary Kullback-Leibler\ndivergence. Throughout the paper, the same Markov/Chebyshev argument\nwill apply every time we will have to deal with a number of clusters with\na specific property.\nWe now compute the total entropy stot =\n1\nN log2 |S|. First note a ran-\ndom configuration belongs on average to 2N(1−α)(1 −p\n2)N clusters. There-\nfore, if\nα < αd := log2 (2 −p),\n(9)\nthen with high probability the total entropy is stot = 1.\nNow assume α > αd. The total entropy is given by a saddle-point\nestimation:\nX\nA\n2s(A)N = [1 + o(1)]N\nZ\nΣ(s)≥0\nds 2N[Σ(s)+s],\n(10)\nwhence\nstot = max\ns\n[Σ(s) + s | Σ(s) ≥0].\n(11)\nWe denote by s∗= argmax[. . . ] the fraction of free variables in the clusters\nthat dominate the sum. Note that in this sum solutions belonging to several\nclusters have been counted too many times. This does not affect the validity\nof our estimation, since in every cluster the fraction of solutions belonging\nto more than one cluster is exponentially small as long as α > αd.\nDefine ̃s := 2(1 −p)/(2 −p) such that ∂sΣ( ̃s) = −1. The complexity of\nclusters with entropy ̃s reads:\nΣ( ̃s) =\np\n2 −p + log2(2 −p) −α.\n(12)\n ̃s maximizes Eq. (11) as long as Σ( ̃s) ≥0, that is if\nα ≤αc :=\np\n(2 −p) + log2 (2 −p).\n(13)\nThen the total entropy reads\nstot = 1 −α + log2 (2 −p)\nfor\nα ≤αc.\n(14)"},{"paragraph_id":"p10","order":10,"text":"6\n-0.05\n0\n0.05\n0.1\n0.15\n0.2\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\ns\nslope −1\nsM\nstot\n-0.01\n0\n0.01\n0.02\n0.03\n0.04\n0.45\n0.5\n0.55\n0.6\n0.65\ns\nslope −1\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.8\n0.85\n0.9\n0.95\n1\n1\n0.8\n0.6\n0.4\n0.2\nΣ∗\nΣtot\nm\nstot = sM\nslope −m\nstot\nαc\nNon-Condensed\nCondensed\ns∗\nΣ∗\ns, Σ\nm\ns∗\ns∗= sM\nΣ(s)\nΣ(s)\nα\nFig. 1 (Color online) Top: graphical construction of the maximum of Σ(s) + s\nby a Legendre transformation. In the top left figure (α < αc), the line of slope\n−1 tangent to the complexity function gives the saddle-point s∗, as well as the\ntotal entropy stot by the intercept on the s axis. In the top right figure (α > αc),\nthe supporting line of slope −1 gets “stuck” at sM, where the derivative is −m >\n−1. Bottom: represented as a function of α: total entropy stot, total complexity\nΣtot = 1 −α, typical entropy s∗, complexity of dominating clusters Σ∗= Σ(s∗),\nand m = −∂sΣ(s∗). The condensation point αc marks the separation between\nthe two regimes illustrated above.\nFor α > αc, the maximum in (11) is realized by the largest possible cluster\nentropy sM, which is given by the largest root of Σ(s). Then stot = s∗=\nsM. In this phase the dominating clusters3 have size eNs∗+∆, where ∆=\nO(1) is asymptotically distributed according to a Poisson point process of\nrate e−m∆, i.e., for d∆≪∆the probability that there is at least one state\nof size between eNs∗+∆and eNs∗+∆+d∆is e−m∆d∆, where m = −∂sΣ(s∗).\nExtreme value study of this process leads to the Poisson-Dirichlet [24,25,26]\ndistribution of weights of clusters. In particular it follows that an arbitrary\nlarge fraction of the solutions can be covered by a finite number of clusters.\nSuch a phase is called condensed.\nIn summary, for a fixed value of the parameter p, and for increasing\nvalues of α, four different phases can be distinguished:\n(a) Liquid phase, α < αd: almost all configurations are solutions.\n3 The “dominating clusters” are the minimal set of clusters covering almost all\nsolutions."},{"paragraph_id":"p11","order":11,"text":"7\n(b) Clustered phase with many states, αd < α < αc: an exponential number\nof clusters is needed to cover almost all solutions.\n(c) Condensed clustered phase, αc < α < αs = 1: a finite number of the\nbiggest clusters cover almost all solutions.\n(d) Unsatisfiable phase, α > αs: no cluster, hence no solution, exists.\nThe very same series of phase transitions is observed in random k-coloring\nand random k-satisfiability, where α is the density of constraints [8,10].\nThe condensation transition at αc corresponds to the Kauzmann temper-\nature [27] in the theory of glasses; at αc the total entropy stot(α) has a\ndiscontinuity in its second derivative with respect to α, in analogy with the\ndiscontinuity of the specific heat at the Kauzmann temperature.\nKeeping in mind the similarity between the RSM and real CSPs, it can\nbe useful to mention some of the properties that are commonly discussed\nin the statistical physics analysis of these problems (for brevity we omit the\nproves of these statements). Among them is the probability distribution of\nmutual overlaps P(q), where q(σ, σ′) = PN\ni=1[2δ(σi, σ′\ni) −1]/N. Below the\ncondensation transition, α < αc, we have P(q) = δ(q), reflecting the fact\nthat random pairs of solutions are uncorrelated. For α > αc, the overlap\nfunction consists of intra-cluster and inter-cluster overlaps: P(q) = wδ[q −\n(1 −stot)] + (1 −w)δ(q), where w is the sum of squares of weights of all\nthe clusters, it is a non-self-averaging random variable, the distribution of\nwhich can be computed from the Poisson-Dirichlet process [26,28].\nAn equivalent way of characterizing the condensed phase is to consider\nthe k-point correlation function, with k ≥2:\nX\nx1,...,xk\n|P(σ1 = x1, σ2 = x2, . . . , σk = xk) −P(σ1 = x1) . . . P(σk = xk)|.\n(15)\nThis quantity decays to zero as N goes to infinity in the non-condensed\nphase, whereas it remains bounded away from zero in the condensed phase.\n2.3 The large k-limit of random k-SAT and k-COL\nOne of the most interesting properties of the random-energy model [19] is\nits equivalence with the large-p limit of the p-spin glass [28]. In the same\nspirit, although the justification is slightly different, the random-subcube\nmodel is found to be equivalent to random k-SAT, random k-COL, and\npresumably other constraint satisfaction problems in the limit k →∞,\nfor connectivities close to the satisfiability threshold. Let us detail this\nstatement.\nIt was already known that at zeroth order (when k →∞) random k-\nSAT and k-COL behave as a random-code model (random-subcube model\nwith p = 1), in which clusters are uniformly distributed singletons. Recent"},{"paragraph_id":"p12","order":12,"text":"8\nlarge-k calculations of the cluster size distribution Σ(s) in k-SAT and k-\nCOL [8,10] allow a direct comparison with the RSM.\nThe control parameters of the RSM are rescaled as:\np = 1 −ε ,\nα = 1 + ε1 + γ\nln 2 ,\n(16)\nwith ε ≪1 and γ = Θ(1). The cluster size distribution in the RSM is then,\nat leading order:\nΣ(s) ln(2) = s\nh\n1 −ln s\nε\ni\n−ε(2 + γ) + o(ε).\n(17)\nwhile the condensation and satisfiability thresholds read, in terms of the\nrescaled variable γ:\nγc = −2 ln 2 ,\nγs = −1.\n(18)\nIn k-SAT and k-COL, identifying ε and γ in the following way:\nSAT :\nε =\n1\n2k+1 ,\nM\nN = 2k ln 2 −ln 2\n2\n+ γ\n2 ,\n(19)\nCOL :\nε = 1\n2k ,\nM\nN = k ln k −ln k\n2\n+ γ\n2 ,\n(20)\nwhere M is the number of constraints (edges in coloring), and N the number\nof variables, gives a perfect match4 for the complexity function (17), as\ncomputed in [8,10]. This illustrates the analogy between α and the density\nof constraints M/N, as well as between p and k.\nThis equivalence goes further than simply having identical cluster size\ndistributions. The cavity analysis of k-SAT shows that the fraction of free\nvariables in a cluster scales exactly as its internal entropy s. The entropy\nof clusters is thus maximal, from which we infer that clusters fill up the\nwhole subcube prescribed by their frozen variables, like in the RSM. The\nsame is true for k-COL (compare Eqs. (E14) and (E27) in [10]) with the\nsmall difference that for every unfrozen variable only two (out of k) colors\nare allowed.\nNote that this comparison is valid only in a finite vicinity of the satis-\nfiability threshold, for γ = Θk(1). In particular, it does not encompass the\nclustering transition, which for k-SAT (resp. k-COL) occurs for constraint\ndensities scaling as 2k ln 2/k (resp. k ln k/2).\n4 Note the difference in the logarithmic base between here and [8,10]."},{"paragraph_id":"p13","order":13,"text":"9\n2.4 Decimation\nAn important contribution of statistical physics to the field of combinatorial\noptimization has been to exploit the information provided by message-\npassing algorithms to devise physics-guided decimation schemes.\nMessage-passing algorithms exchange information between units (vari-\nables and constraints) in order to obtain estimates of marginal probabilities\n(beliefs) or other related quantities (e.g. surveys, see below). Subsequently,\nthis information is used to find a solution. A usual way to do this, called\ndecimation, proceeds as follows: fix randomly the value of one5 variable ac-\ncording to its estimated belief, then re-run the message-passing algorithm\non the reduced system, and loop. A trivial statement is that a perfect\nestimate of all marginal probabilities would always cause the decimation\nprocedure to find a solution (if any).\nIn the RSM there is no underlying graph, therefore message-passing can-\nnot be defined. However, it is possible to study decimation schemes based\non exact marginal probability estimators. Although such procedures should\nreally be viewed as thought experiments, they can be used to gain some\ninsight on real algorithms. Here two idealized algorithms are considered:\n– Belief estimator: outputs the exact marginal probabilities μi(σi) =\nP\nσ\\i μ(σ), where μ(σ) = I(σ ∈S)/|S|.\n– Survey estimator: outputs “surveys”, i.e. marginal probabilities over the\nclusters: νi(πi) = P\nπ\\i ν(π), where ν(π) = P\nA I(π = πA)/⌊2N(1−α)⌋.\nIn real CSPs, belief and survey propagation arguably provide asymp-\ntotically accurate estimators, as long as the number of clusters dominating\nthe measure μ (or ν for survey propagation) scales exponentially with N,\nand as long as the one-step replica symmetry breaking description is the\ncorrect one [14,10]. However, belief-guided and survey-guided decimation\nschemes are difficult to analyze in real CSPs — see [29] for an empirical\nstudy on surveys and [30] for recent analytical study on beliefs in k-SAT.\nLet us study decimation in the RSM. As long as the phase is non-\ncondensed, the belief estimator always outputs μi(σi) ≈1/2 for all i in\nthe limit N →∞. Likewise, the survey estimator will output νi({0}) ≈\nνi({1}) ≈p/2, νi({0, 1}) ≈1 −p. In both cases, the decimation procedure\nis completely unbiased: it will fix a random variable i to 0 or 1, with proba-\nbility 1/2. This observation remains true in the subsequent decimation steps\nas long as the number of clusters dominating the reduced measure μ (or\nν) remains exponential. Within this assumption, after T = tN (0 ≤t ≤1)\ndecimation steps, T variables will be fixed randomly and independently.\nWe are then left with a restricted space of solutions compatible with these\nT fixed variables. The logarithm of the number of clusters of entropy sN\n5 In practice the number of variables fixed at each step can range from one to\na small fraction of the variables."},{"paragraph_id":"p14","order":14,"text":"10\nis then\nNΣt(s) = N"},{"paragraph_id":"p15","order":15,"text":"1 −α + t log2"},{"paragraph_id":"p16","order":16,"text":"1 −p\n2"},{"paragraph_id":"p17","order":17,"text":"−(1 −t)D"},{"paragraph_id":"p18","order":18,"text":"s\n1 −t||1 −p"},{"paragraph_id":"p19","order":19,"text":".\n(21)\nRescaling by the number of unfixed variables (1 −t)N: Σt = (1 −t) ̄Σt,\ns = (1 −t) ̄s, we obtain\n ̄Σt( ̄s) = 1 −α −tαd\n1 −t\n−D( ̄s||1 −p).\n(22)\nThe parameter ̄α(t) := (α −tαd)/(1 −t) now plays the same role as α\nin the analysis of the RSM. Consequently, the system undergoes the same\ncondensation and unsatisfiability transitions as t is increased. Assume for\nexample that αd < α < αc. Fixing a fraction tc := (αc −α)/(αc −αd)\nof the variables will cause the system to condense. The belief estimator\nwill then be dominated by a finite number of clusters, yielding instance-\ndependent biases on variables. An extensive number of variables become\nsuddenly near-frozen, i.e., μi(1) ≈0 or ≈1, and remain so for t > tc. At\nts := (1 −α)/(1 −αd), the total complexity goes to zero, and near-frozen\nvariables become truly frozen, i.e. μi(1) = 0 or 1, as all sub-dominant\nclusters disappear. By contrast, the survey estimator will output unbiased\nmarginal probabilities as long as the total number of clusters is exponential\nin N, that is if t < ts. At t = ts, decimation concentrates on a single\ncluster, causing a freezing avalanche in the surveys, i.e. for each variable\ni, νi({0, 1}) = 1 or νi({0}) = 1 or νi({1}) = 1. For both estimators, any\nslight error at ts will cause the failure of the decimation process.\nNote that in real CSPs, belief propagation is not expected to be correct\nin the condensed phase, beyond tc; in fact, it will not detect condensation,\nnor near-frozen variables. For a recent study of the belief-propagation-\nguided decimation in k-SAT see [30]. Remarkably, in real CSPs, survey-\nguided decimation usually simplifies the problem: after a certain number\nof decimation steps, it outputs νi({0, 1}) = 1 for all i, and the problem can\nthen be passed over to a simple local search algorithm [6,29]. Although the\nRSM, which is intrinsically clustered, is unable to capture this property, it\nsheds some light on why decimation may not work in some cases.\n3 Distance and ergodicity\nThe geometrical organization of solutions to CSPs is thought to play an\nimportant role in setting intrinsic limits to the performance of search al-\ngorithms. In particular, the clustering phenomenon, by which the solution\nspace is fragmented into many connected components far from each other,\nhas been proposed by physicists as a possible explanation for the failure of\nmost known algorithms [4,5,6]. Also, the role of frozen variables was re-\ncently discussed in [10]. Conversely, the success of survey propagation [6] is"},{"paragraph_id":"p20","order":20,"text":"11\nusually explained by the fact that it explicitly incorporates the existence of\nclusters. The separability of clusters has been proved in the k-SAT problem,\nin compliance with the predictions of statistical physics [15]. Despite this\nevidence, the precise relation between geometry and algorithms still lacks a\nrigorous base. The random-subcube model offers an excellent opportunity\nto study these questions in a well controlled framework.\n3.1 The dynamical transition\nLet us first argue why the clustering transition at αd, defined by Eq. (9),\nactually corresponds to what is commonly refered to as the dynamical\n(ergodicity breaking) transition in real CSPs [31,32,33,8,12,10]. To this\nend we study a uniform unbiased random walk on the space of solutions:\nat each step, one is allowed to move from one solution to the other by\nflipping only one spin. We choose this dynamics for the sake of simplicity,\nbut most of the arguments below hold for more general dynamical rules,\nlike for example the flipping of a sub-extensive number of spins at each\nstep.\nWe have already pointed out that an arbitrary configuration belongs\nto 2(1−α)N(1 −p/2)N different clusters w.h.p. if α < αd, and to none if\nα > αd. Therefore below αd almost all configurations are solutions, and\nany reasonable dynamics will explore the entire phase space uniformly.\nOn the other hand, when α > αd, solutions become exponentially rare.\nLet A be a cluster of internal entropy s. What is the probability that after t\nsteps, a random walker ends up in another cluster B of internal entropy s′?\nLet a denote the proportion of variables that are free in A and frozen in B.\nThe probability distribution of a is given by the total number of partitions\nof {1, . . . , N} into four categories: frozen in A and B, only frozen in A, only\nfrozen in B, and frozen neither in A nor B. This probability reads:\nq(a) =\n1\n N\nNs\n N\nNs′"},{"paragraph_id":"p21","order":21,"text":"N!\n(Na)![N(s −a)]![N(1 −s′ −a)]![N(s′ −s + a)]!.\n(23)\nIn order for the walker to reach B from A, it has to match perfectly the\nprescriptions (freezings) of B on these aN variables. If t = Θ(N d), the prob-\nability of this happening is ≈t2−aN. Additionally, variables that are frozen\nin both clusters must coincide, so that A and B have a non-empty intersec-\ntion. For a random choice of B, this happens with probability 2N(s′+a−1).\nConsequently, the probability that the walker wanders in any other\ncluster of entropy s′ after t steps is union-bounded by:\nφ(s →s′) ≤2NΣ(s′) X\na\nq(a) t 2−aN 2N(s′+a−1)\n(24)"},{"paragraph_id":"p22","order":22,"text":"12\nThe maximum of\n1\nN log q(a) tends to zero when N →∞, so that:\nlim sup\nN→∞\n1\nN log φ(s →s′) ≤Σ(s′) + s′ −1\n(25)\nThis quantity remains negative for all s′, as long as α > αd. Therefore,\nin this regime, hopping from one cluster to the other is very unlikely in\na sub-exponential number of steps, even though clusters are not disjoint.\nIn this sense we say that ergodicity is broken. Remarkably, the space of\nsolutions becomes non-ergodic as soon as it becomes non-trivial, at αd.\nLet us stress the importance of the entropic barriers between clusters\nin our analysis. In the real CSPs and optimization problems the energetic\nbarriers are usually thought of as more important, and clusters are even\nsometimes described as separated (by an extensive distance). The impor-\ntance of entropic barriers and the possibility of non-extensively separated\nclusters should, however, not be neglected in the studies of richer models.\n3.2 x-satisfiability\nThe notion of x-satisfiability was first introduced as a tool to study the\ngeometrical structure of the solution space of CSPs [15]. An instance of CSP\nis said x-satisfiable if and only if it admits a pair of solutions separated by a\nHamming distance ∼xN. In other words, x-satisfiability gives the distance\nspectrum of the solution space. This spectrum is estimated using three\nquantities:\na) d1 = x1(α)N: the maximum distance between two solutions inside one\ncluster,\nb) d2 = x2(α)N: the minimum distance between two solutions from two\ndistinct clusters,\nc) d3 = x3(α)N: the maximum distance between any two solutions (pre-\nsumably from two different clusters).\nThe first of these quantities is estimated by noting that the maximum\ndistance between any two solutions in a given cluster, i.e. its diameter,\nequals its entropy s. Therefore the maximum diameter/entropy x1 is given\nw.h.p. by the largest internal entropy sM, i.e. the largest root of Σ(s) =\n1 −α −D(s ∥1 −p).\nNow take two clusters A and B at random, and consider the probability\nthat their distance be xN. This distance is given by the number of variables\nwhich are frozen in both clusters, but in a contradictory way, such that\nπA(i) ̸= πB(i). This happens independently with probability p2/2 for each\nvariable, so that the number of such variables follows a binomial law of\nparameter p2/2. Therefore, the number N(x) of pairs of clusters at distance"},{"paragraph_id":"p23","order":23,"text":"13\nαs(x)\nx1\nx2\nαgap\nαsep\nαc\nx\nα\n1\n0.9\n0.8\n0.7\n0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0\n1\n0.9\n0.8\n0.7\n0.6\nFig. 2 (Color online) The x-satisfiability threshold is constructed from the three\ndistances x1(α), x2(α) and x3(α). Below a threshold αgap, distance spectra fail to\ndetect the fragmentation of the solution space (there is no more “gap” between\nintra and inter-cluster distances). Below another threshold αsep, clusters cease\nto be all well separated, although ergodicity is still broken. The condensation\nthreshold αc is shown for information. The dynamical threshold αd lies outside\nthe picture, and its value is ≈0.070. In this figure p = 0.95.\nxN coincides w.h.p. with its mean value:\nE[N(x)] = 22(1−α)N\n N\nNx"},{"paragraph_id":"p24","order":24,"text":"1 −p2\n2\n (1−x)N p2\n2\n xN\n≍2Ns2(x)\n(26)\nif s2(x) := 2(1 −α) −D(x ∥p2/2) > 0, and N(x) = 0 w.h.p. if s2(x) < 0.\nConsequently the smallest possible distance between any two clusters is\ngiven by x2N, where x2(α) is the smallest root of s2(x). A similar argument\ngives the largest distance between any two solutions from two different\nclusters: x3(α) = 1 −x2(α).\nTo sum up, we find that a random instance is x-satisfiable w.h.p. if\nα < αs(x), and is x-unsatisfiable w.h.p. if α > αs(x), with:\nαs(x) ="},{"paragraph_id":"p25","order":25,"text":"1\nif x ∈[0, 1 −p] ∪[p2/2, 1 −p2/2]\n1 −D(x ∥1 −p)\nif x ∈[1 −p, x0]\n1 −1\n2D(x ∥p2/2)\nif x ∈[x0, p2/2]\n1 −1\n2D(1 −x ∥p2/2) if x ∈[1 −p2/2, 1]\n(27)\nwhere x0 is solution to D(x ∥p2/2) = 2D(x ∥1 −p)."},{"paragraph_id":"p26","order":26,"text":"14\nFig. 2 shows how αs(x) can be constructed from the three distances x1,\nx2, x3. We put αsep := 1 + (1/2) log(1 −p2/2) > αd, the threshold below\nwhich some pairs of clusters have a non-empty intersection. An interesting\nobservation is that ergodicity can still be broken even below this threshold.\nWe can also define the αgap := αs(x0) > αsep, below which distances\nfrom the same cluster and distances from distinct clusters overlap. This\nthreshold sets the limit below which the notion of x-satisfiability fails to\ndetect clustering. The random-subcube model allows us to make a clear\nand intelligible distinction between the three thresholds αd < αsep < αgap.\nWe expect this distinction to hold in most CSPs6.\n4 Random energy landscape\n4.1 Definition\nThe random-subcube model can be enriched by adding the notion of energy\nto the definition of states. The motivation for doing this is to mimic the\noptimization version of CSPs (where energy is defined as the number of un-\nsatisfied constraints), but it can also be used to reproduce some properties\nof glassy systems.\nFor each energy level E0, we define N(E0) = ⌊2NΣ(E0/N)⌋valleys of\nenergy E0, where Σ(e0) is an increasing complexity function. This function\ncan be arbitrary, but for simplicity we will restrict our examples to the\nform7 Σ(e0) = a + be0 −ce0 ln(e0), where a > 0 corresponds to a SAT\nphase, and a < 0 to an UNSAT phase.\nEach valley is defined as a subcube V , chosen at random in the same\nway as clusters in the previous sections, cf. (1). In the following, the freezing\nprobability p will be fixed for all energies, but one could easily generalize\nthe model by making it energy-dependent: p = p(e0).\nThe number N(E0, S0) of valleys of energy E0 = e0N and entropy\nS0 = s0N is w.h.p.:\nN(E0, S0) ≍2NΣ(e0,s0) if Σ(e0, s0) := Σ(e0) −D(s0 ∥1 −p) ≥0\n= 0\notherwise.\n(28)\nGiven a configuration σ, we define its energy as a trade-offbetween the\nenergy of surrounding valleys and their distance. Let us denote the energy\nof a valley V by E0(V ). Then the energy of σ is:\nE(σ) := min\nV\n[E0(V ) + d(σ, V )]\n(29)\n6 With the notable exception of k-XORSAT, where αd = αsep. Incidentally in\nk-XORSAT we also have αc = αs.\n7 This form corresponds to the function used to fit data from the cavity method\nin the k-SAT problem [6]."},{"paragraph_id":"p27","order":27,"text":"15\nwhere d(σ, V ) is the distance between σ and the nearest element of V .\nBy definition, we say that σ belongs to the basin of attraction of V if\nV minimizes the sum. Observe that with this definition, it may happen\nthat some valleys are not represented at all in the energy landscape. In the\nfollowing, the term “state-bottom energy” shall refer to the energy E0 of\nthe valley minimizing the sum, and the term “state-bottom entropy” to the\nentropy S0 of that valley.\n4.2 Static description of the energy landscape\nWhat is the energy of an arbitrary configuration σ? Let us start with\nthe typical case: for each e0 = E0/N, we compute the distance to the\nnearest valley with energy E0. Standard arguments show that the number\nof valleys of state-bottom energy e0N at distance d = ωN is governed by\nthe exponent:\nΣ(e0) −D(ω ∥p/2).\nThen, the minimum distance is given, in the N\n→\n∞limit, by\nδ[Σ(e0), p/2]N, where δ(x, y) is solution to: x = D(δ ∥y). Then, the typi-\ncal energy is obtained as the best compromise between state-bottom energy\nand distance:\ne∗= E/N = min\ne0 {e0 + δ[Σ(e0), p/2]} ,\n(30)\nThe argmin gives the typical state-bottom energy e∗\n0 of a random σ.\nAs we just saw, most configurations have roughly the same energy, and\nbelong to valleys with the same state-bottom energy. At finite temperature\nhowever, thermodynamics will be dominated by configurations of lower\nenergy than e∗N. We thus need to estimate the entropy function (governing\nthe number of configurations of given energy), the Legendre transform of\nwhich shall give us the free energy. Given a valley V of energy E0 = e0N\nand entropy S0 = s0N, the number of configurations of energy E belonging\nto this valley is:\nNV (E) = 2S0\n E −E0\nN −S0"},{"paragraph_id":"p28","order":28,"text":",\n(31)\n≍2NsV (e|e0,s0),\nwith\nsV (e|e0, s0) := s0 + (1 −s0)H\n e −e0\n1 −s0"},{"paragraph_id":"p29","order":29,"text":",\nwhere H(x) = −x log2 x−(1−x) log2 (1 −x) is the entropy function. Sum-\nming up over all valleys, the total number of configurations with energy\nE = eN is:\nN(E) =\nX\nS0,E0\n2S0\n E −E0\nN −S0"},{"paragraph_id":"p30","order":30,"text":"2NΣ(E0/N,S0/N) ≍2Ns(e),\n(32)\nwith\ns(e) =\nmax\ne0,s0\nΣ(e0,s0)≥0\n[sV (e|e0, s0) + Σ(e0, s0)] .\n(33)"},{"paragraph_id":"p31","order":31,"text":"16\nHere we have implicitly assumed that all elements in the sphere of radius\nE−E0 and center V are in the basin of attraction of V , as long as E < e∗N.\nThis is not true in general, as some configurations may in fact belong to\na more favorable basin, and may thus have lower energies. However, such\nconfigurations remain exponentially rare in comparison to the total weight\nof the sphere. Therefore the previous estimate holds8.\nCanceling the derivative w.r.t. s0 in (33) yields the saddle for s0:\n ̃s0 = (1 −p)(1 −e + e0)\n1 −p/2\n(34)\nProvided that the maximum is reached in a region where Σ(e0, ̃s0) > 0, we\nget:\ns(e) = max\ne0 [1 −D(e −e0 ∥p/2) + Σ(e0)]\nfor\nec < e < e∗.\n(35)\nThis will be valid from e = e∗(for which we find s(e∗) = 1 as expected)\ndown to a certain condensation energy ec. Below that energy (e ≤ec), the\nphase is condensed: Σ(e0, ̃s0) < 0, and the maximum in (33) is reached on\nthe border on the definition domain, where Σ(e0, s0) = 0. If we denote by\nsM(e0) the biggest valley of energy e0 (i.e. the largest of root of Σ(e0, s0) =\n0), we get:\ns(e) = max\ne0 {sV [e|e0, sM(e0)]}\nfor\ne < ec.\n(36)\nSo far we have worked in the microcanonical ensemble, but the same ar-\nguments hold in the canonical ensemble. In particular the condensation\ntemperature is Tc = (∂s/∂e|e=ec)−1. The number of dominating states in\nthe condensed phase follows again a Poisson-Dirichlet process [25,26] with\nparameter m, where m/T is the slope of the curve ΣT (f) at its smallest\nroot. The function ΣT (f) is the canonical counterpart of Σ(e0, s0), and\nreads:\nΣT (f) =\nmax\ne0,s0:fV (T |e0,s0)=f Σ(e0, s0) ,\n(37)\nwhere fV (T |e0, s0) is the single-state free energy, obtained as the Legendre\ntransform of sV (e|e0, s0) in (31).\nIn both low-temperature phases (condensed and non-condensed), equi-\nlibrium is reached for different values of (e0, s0) as e varies. Said differently,\nthe states dominating the microcanonical measure at e and at e + δe are\ncompletely distinct. In the canonical language, we say that the system ex-\nhibits temperature chaos [34,35]: slightly changing the temperature from T\nto T + δT dramatically modifies the free-energy landscape, reshuffling the\nordering of states. Consequently, correlations are nonexistent between T\n8 This argument is similar to the one used in Eq. (10), where some solutions\nwere counted several times, but with no consequence at the exponential scale."},{"paragraph_id":"p32","order":32,"text":"17\n 0.8\n 0.82\n 0.84\n 0.86\n 0.88\n 0.9\n 0.92\n 0.94\n 0.96\n 0.98\n 1\n 0.03\n 0.04\n 0.05\n 0.06\n 0.07\n 0.08\nV3\nV2\nV1\nsV2(e)\nsV3(e)\nslope 1/T\nsV1(e)\nsM(e0)\ns\ne\nFig. 3 (Color online) Illustration of the temperature chaos by construction of the\nmicrocanonical entropy in the condensed phase, cf. Eq. (36), and level crossing.\nWe have represented three entropy curves sV (e) corresponding to three extremal\nstates V1, V2 and V3, the envelope of all these curves is the microcanonical entropy.\nThese states are maximally atypical; they realize a balance between low state-\nbottom energies and a high state-bottom entropies, which are related by s0 =\nsM(e0) (thick curve). As the temperature (or energy) is decreased, the curves\nsV (e) cross each other, and the system is dominated by states of lower state-\nbottom energies and entropies. These data were obtained for number of valleys\nΣ(e0) = −0.05 −0.5e0 ln e0 and p = 0.2.\nand T +δT . This phenomenon of free-energy crossings is illustrated (in the\ncondensed phase) by Fig. 3, where the maximum of Eq. (36) is constructed\ngeometrically.\n4.3 Relation with dynamics\nWe now undertake to describe the dynamical properties of this energetic\nlandscape. To that end we shall make use of the static picture, which we\nknow precisely from bottom-up construction.\nOur reasoning proceeds in two steps. First, we study the behavior of\na single spin-flip Monte-Carlo dynamics with detailed balance evolving in\na single valley of state-bottom energy e∗\n0, and state-bottom entropy s∗\n0. In\na second step, we argue that the same dynamics run on the full rugged\nenergy landscape is entirely governed by this single-state-like behavior.\nThe thermodynamics of a single typical state is given by:\ns(e|e∗\n0, s∗\n0) = s∗\n0 + (1 −s∗\n0)H\n e −e∗\n0\n1 −s∗\n0"},{"paragraph_id":"p33","order":33,"text":"(38)"},{"paragraph_id":"p34","order":34,"text":"18\nwhere s∗\n0 = (1−p)(1−e∗+e∗\n0)/(1−p/2) is the typical state-bottom entropy\nof a random configuration, computed from Eq. (34). Supposing now that the\nenergy landscape is made of this state only, we can easily convince ourselves\nthat the dynamics is ergodic: the energy landscape is convex, and there are\nneither energetic nor entropic barriers. Both the quenched and annealed9\ndynamics in this simple landscape is then given by the thermodynamics.\nWe know from the previous paragraphs that states extend up to energy\ne∗, where a “crest” connects the different valleys. This crest is actually more\nof a plateau, as it embeds almost all configurations: s(e∗) = 1. Therefore,\nour dynamics will remain there as long as the temperature does not al-\nlow configurations of lower energy. This happens at Td := (∂s/∂e|e=e∗)−1,\nwhere exploring valleys starts to be more favorable.\nBelow Td, the system will find itself trapped in one state, since barri-\ners between valleys are extensive. This randomly picked state has typical\nproperties: in particular, its bottom-energy is e∗\n0 and its bottom-entropy is\ns∗\n0.\nWe argue that below Td everything happens as if this trapping state was\nput in isolation, as we have described above. The justification comes from\nthe fact that a valley does not “see” its neighbors as long as e < e∗: even\nthough many configurations of energy e < e∗in the isolated state actually\nbelong other valleys in the full landscape, their proportion is exponentially\nsmall. In other words, although there may be some directions for which the\nenergy barrier is lower than e∗−e, these directions are beaten entropically.\nWith this reasoning, both the quenched and annealed dynamics are\ndescribed by the microcanonical entropy of a single typical state: sdyn(e) =\ns(e|e∗\n0, s∗\n0). We note in passing:\nT −1\nd\n= ∂s\n∂e"},{"paragraph_id":"p35","order":35,"text":"e=e∗= ∂sdyn\n∂e"},{"paragraph_id":"p36","order":36,"text":"e=e∗,\n(39)\nas partial derivatives w.r.t. e0 and s inside (33) cancel at the maximum.\nThe dynamical temperature Td is thus well defined.\nNote that this analysis is exact: the fact that it purely relies on static\narguments should be attributed to the simplicity of the model. In richer\nmean-field models this kind of arguments might also apply, but will most\nprobably not give the full picture.\n4.4 Glassy behaviour\nThe analysis of the energy landscape in the RSM is summarized in the\nenergy-temperature phase diagram of Fig. 4. As anticipated, the behaviour\n9 By quench, resp. annealing, we mean a fast, resp. slow, change of temperature\nfrom T1 > Td to T2 < Td and some time (a finite number of Monte Carlo sweeps)\nspent at T2 afterwards."},{"paragraph_id":"p37","order":37,"text":"19\n0\n0.5\n1\nm\n0\n0.2\n0.4\n0.6\n0.8\n1\n1.2\n1.4\nT\n0\n0.2\n0.4\n0.6\n0.8\n1\n1.2\n1.4\nT\n0\n0.05\n0.1\n0.15\n0.2\n0.25\nenergy\ne∗\nec\negs\ne∗\n0\nTd\nTc\ncondensed glass\nglass\nliquid\ndynamical\nFig. 4 (Color online) Energy as a function of temperature. At temperature\nT > Td, the system is in a liquid state: the dynamics is exploring ergodically\nall configurations of energy e∗. Below the dynamical temperature ergodicity is\nbroken. The upper “dynamical” curve shows the result of a quench/annealing\nin temperature, whereby the systems remains stuck in a typical state of bot-\ntom energy e∗\n0. For Tc < T < Td, equilibrium thermodynamics is dominated\nby an exponential number of states (curve “glass”). Below the condensation\n(Kauzmann) temperature thermodynamics is dominated by a finite number of\nstates (curve “condensed glass”). This number is given by a Poisson-Dirichlet\nprocess of parameter m (plotted in the upper part of the diagram). The dashed\nline shows the result of a quench/annealing starting from an equilibrium state\nat temperature Tc < T < Td. The bottom line shows the bottom energies\nof the thermodynamically dominating states. These curves were obtained for\nΣ(e0) = −0.05 −0.5e0 ln e0 and p = 0.6.\nof the RSM resembles the one observed in glasses and spin-glasses. The\ntwo distinct glassy transitions (dynamical and condensation), as well as\nthe phenomenon whereby the physical dynamics gets stuck in metastable\nstates, have been described for example in the p-spin glass [28], the spherical\np-spin glass [36], the Potts glass [37] and the lattice glass [38]. Several\nrelated examples of energy-temperature diagrams were derived recently in\n[39].\nIn the aforementioned mean-field models, the static behaviour is bet-\nter understood than the dynamics. Static properties are usually analyzed\nby the replica/cavity method, with the help of Parisi’s replica symmetry\nbreaking scheme. A satisfactory analytic treatment of the dynamics exists"},{"paragraph_id":"p38","order":38,"text":"20\nonly for the spherical p-spin glass [36], in which all states have the same\nentropy. A remarkable step towards connecting the static picture and dy-\nnamical behaviour in a rather general framework was done in [31,32,33].\nThe RSM could provide a tractable playground for studying several aspects\nof glassy dynamics, e.g. aging and rejuvenation [40].\nWe would like to emphasize the freedom we can enjoy in the definition\nof the energy landscape. First, arbitrary numbers and sizes of valleys at\neach energy e0, Σ(e0) and p(e0), can be considered. Second, the definition\nof the configurational energy E(σ) in Eq. (29) could be generalized to an\narbitrary function of all valleys V and σ; E(σ) = F({V }, σ). By tun-\ning these parameters, one could hope to reproduce the dynamics of more\ncomplex models on a quantitative level.\n5 Conclusions\nThe random-subcube model is a simple exactly solvable model capturing\nseveral interesting properties of random constraint satisfaction problems.\nRather than an attempt to construct a new realistic model for practical\ninstances of constraint satisfaction problems, it allows us to identify which\nproperties of random CSPs can be reproduced by a simple probabilistic\nstructure, and conversely, which of these properties may be intrinsically\nnon-trivial. Examples of reproducible properties include condensation, non-\nmonotony of the x-satisfiability threshold, temperature chaos and dynami-\ncal freezing in metastable states. From this point of view, the RSM stands\njust next to the random-energy model [19], the random-code model [20,21,\n22] or the random-energy random-entropy model [35].\nSince the relation between the RSM and the large-k limit of random\nk-SAT and k-COL is based on non-rigorous results from [8,10], it would\nbe interesting to establish this equivalence rigorously. Further, the RSM\nshould be helpful for understanding some properties which are too difficult\nto study in more realistic models, such as finite-size corrections or some\naspects of glassy dynamics.\nFinally, our work addresses the broad question of producing, infer-\nring, and representing complex and rugged structures of the hypercube.\nAlthough we retained the simplest choice of subcubes for clusters, more so-\nphisticated alternatives could be explored and used to reproduce detailed\ngeometrical features of solution spaces in CSPs.\n6 Acknowledgment\nWe would like to thank Florent Krz ֒aka la and Marc M ́ezard for fruitful\ndiscussions about this work, and Guilhem Semerjian and Jiˇr ́ı ˇCern ́y for\ncritical reading of the manuscript. This work has been partially supported\nby EVERGROW (EU consortium FP6 IST)."},{"paragraph_id":"p39","order":39,"text":"21\nReferences\n1. C. H. Papadimitriou. Computational complexity. Addison-Wesley, 1994.\n2. S. Kirkpatrick and B. Selman. Critical behavior in the satisfiability of random\nboolean expression. Science, 264:1297–1301, 1994.\n3. R. Monasson, R. Zecchina, S. Kirkpatrick, B. Selman, and L. Troyansky.\nDetermining computational complexity from characteristic phase transitions.\nNature, 400:133–137, 1999.\n4. G. Biroli, R. Monasson, and M. Weigt.\nA variational description of the\nground state structure in random satisfiability problems. Eur. Phys. J. B,\n14:551, 2000.\n5. M. M ́ezard, G. Parisi, and R. Zecchina. Analytic and algorithmic solution of\nrandom satisfiability problems. Science, 297:812–815, 2002.\n6. M. M ́ezard and R. Zecchina.\nRandom k-satisfiability problem: From an\nanalytic solution to an efficient algorithm. Phys. Rev. E, 66:056126, 2002.\n7. F. R. Kschischang, B. Frey, and H.-A. Loeliger. Factor graphs and the sum-\nproduct algorithm. IEEE Trans. Inform. Theory, 47(2):498–519, 2001.\n8. Florent Krzakala, Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Se-\nmerjian, and Lenka Zdeborov ́a.\nGibbs states and the set of solutions of\nrandom constraint satisfaction problems. Proc. Natl. Acad. Sci., 104:10318,\n2007.\n9. M. M ́ezard, M. Palassini, and O. Rivoire. Landscape of solutions in constraint\nsatisfaction problems. Phys. Rev. Lett., 95:200202, 2005.\n10. L. Zdeborov ́a and F. Krzakala. Phase transitions in the coloring of random\ngraphs. Phys. Rev. E, 76:031131, 2007.\n11. S. Cocco, O. Dubois, J. Mandler, and R. Monasson. Rigorous decimation-\nbased construction of ground pure states for spin glass models on random\nlattices. Phys. Rev. Lett., 90:047205, 2003.\n12. M. M ́ezard, F. Ricci-Tersenghi, and R. Zecchina. Alternative solutions to\ndiluted p-spin models and XORSAT problems. J. Stat. Phys., 111:505, 2003.\n13. T. Mora and M. M ́ezard. Geometrical organization of solutions to random\nlinear Boolean equations. Journal of Statistical Mechanics: Theory and Ex-\nperiment, 10:P10007, October 2006.\n14. M. M ́ezard and G. Parisi. The bethe lattice spin glass revisited. Eur. Phys.\nJ. B, 20:217, 2001.\n15. M. M ́ezard, T. Mora, and R. Zecchina. Clustering of solutions in the random\nsatisfiability problem. Physical Review Letters, 94:197205, 2005.\n16. Dimitris Achlioptas and Federico Ricci-Tersenghi. On the solution-space ge-\nometry of random constraint satisfaction problems. In STOC ’06: Proceedings\nof the thirty-eighth annual ACM symposium on Theory of computing, pages\n130–139, New York, NY, USA, 2006. ACM Press.\n17. A. Montanari and D. Shah. Counting good truth assignments of random k-\nsat formulae. In Proceedings of the 18th Annual ACM-SIAM Symposium on\nDiscrete Algorithms, pages 1255–1264, New York, USA, 2007. ACM Press.\n18. O. Dubois and J. Mandler. The 3-xorsat threshold. FOCS, 00:769, 2002.\n19. B. Derrida. Random-energy model: Limit of a family of disordered models.\nPhys. Rev. Lett, 45:79–82, 1980.\n20. C. E. Shannon. A mathematical theory of communication. Bell System Tech.\nJournal, 27:379–423, 623–655, 1948.\n21. A. Montanari. The glassy phase of Gallager codes. Eur. Phys. J. B., 23:121–\n136, 2001.\n22. A. Barg and G. D. Forney Jr. Random codes : minimum distances and error\nexponents. IEEE Trans. Inform. Theory, 48:2568–2573, 2002.\n23. Guilhem Semerjian. On the freezing of variables in random constraint sat-\nisfaction problems.\nJ. Stat. Phys., 130:251, 2008.\n(to appear) preprint\narXiv.org:0705.2147."},{"paragraph_id":"p40","order":40,"text":"22\n24. M. M ́ezard, G. Parisi, N. Sourlas, G. Toulouse, and M. A. Virasoro. Replica\nsymmetry breaking and the nature of the spin-glass phase.\nJ. Physique,\n45:843–854, 1984.\n25. M. Talagrand. Rigorous low temperature results for the p-spin mean field\nspin glass model. Probability Theory and Related Fields, 117:303–360, 2000.\n26. J. Pitman and M. Yor.\nThe two-parameter poisson-dirichlet distribution\nderived from a stable subordinator. Ann. Probab., 25:855–900, 1997.\n27. W. Kauzmann. The nature of the glassy state and the behavior of liquids at\nlow temperatures. Chem. Rev., 43:219, 1948.\n28. D.J. Gross and M. M ́ezard. The simplest spin glass. Nucl. Phys. B, 240:431,\n1984.\n29. G. Parisi.\nSome remarks on the survey decimation algorithm for k-\nsatisfiability. arXiv:cs/0301015, 2003.\n30. A. Montanari, F. Ricci-Tersenghi, and G. Semerjian.\nSolving con-\nstraint satisfaction problems through belief propagation-guided decimation.\narXiv:0709.1667v1 [cs.AI], 2007.\n31. A. Montanari and G. Semerjian. From large scale rearrangements to mode\ncoupling phenomenology. Phys. Rev. Lett., 94:247201, 2005.\n32. A. Montanari and G. Semerjian. On the dynamics of the glass transition on\nbethe lattices. J. Stat. Phys., 124:103–189, 2006.\n33. A. Montanari and G. Semerjian. Rigorous inequalities between length and\ntime scales in glassy systems. J. Stat. Phys., 125:23, 2006.\n34. A. J. Bray and M. A. Moore. Chaotic nature of the spin-glass phase. Phys.\nRev. Lett., 58:57–60, 1987.\n35. F. Krzakala and O. C. Martin. Chaotic temperature dependence in a model\nof spin glasses. Eur. Phys. J. B., 28:199–208, 2002.\n36. L. F. Cugliandolo and J. Kurchan. Analytical solution of the off-equilibrium\ndynamics of a long-range spin glass model. Phys. Rev. Lett., 71:173, 1993.\n37. D. J. Gross, I. Kanter, and H. Sompolinsky. Mean-field theory of the potts\nglass. Phys. Rev. Lett., 55(3):304–307, Jul 1985.\n38. G. Biroli and M. M ́ezard. Lattice glass models. Phys. Rev. Lett., 88:025501,\n2002.\n39. F. Krzakala and J. Kurchan. A landscape analysis of constraint satisfaction\nproblems. Phys. Rev. B, 76:021122, 2007.\n40. J.-L. Barrat, M.V. Feigelman, J. Kurchan, and J. Dalibard. Slow Relaxations\nand Nonequilibrium Dynamics in Condensed Matter Les Houches Session\nLXXVII, 1-26 July, 2002. Springer, Berlin, 2003."}],"pages":[{"page":1,"text":"arXiv:0710.3804v2 [cs.CC] 28 Jan 2008\nNoname manuscript No.\n(will be inserted by the editor)\nThierry Mora · Lenka Zdeborov ́a\nRandom subcubes as a toy model for\nconstraint satisfaction problems\nReceived: date / Accepted: date\nAbstract We present an exactly solvable random-subcube model inspired\nby the structure of hard constraint satisfaction and optimization problems.\nOur model reproduces the structure of the solution space of the random k-\nsatisfiability and k-coloring problems, and undergoes the same phase tran-\nsitions as these problems. The comparison becomes quantitative in the\nlarge-k limit. Distance properties, as well the x-satisfiability threshold, are\nstudied. The model is also generalized to define a continuous energy land-\nscape useful for studying several aspects of glassy dynamics.\nKeywords Constraint satisfaction problems · Clustering of solutions ·\nExactly solvable models\n1 Introduction\nCombinatorial optimization and Constraint Satisfaction Problems (CSPs)\narise in a wide array of scientific branches, including statistical physics,\ninformation theory, inference and machine learning. These problems, which\ninvolve a large number of variables interacting through a large number of\nconstraints or cost terms, are in general very hard to solve, and in most\nT. Mora\nLewis Sigler Institute for Integrative Genomics, Princeton University, Princeton,\nNJ 08544, USA\nL. Zdeborov ́a\nUniversit ́e Paris-Sud, LPTMS, UMR8626, Bˆat. 100, Universit ́e Paris-Sud 91405\nOrsay cedex, France\nCNRS, LPTMS, UMR8626, Bˆat. 100, Universit ́e Paris-Sud 91405 Orsay cedex,\nFrance"},{"page":2,"text":"2\ncases no algorithm seems to be able to find a solution within a reasonable\ntime, as formalized by the P ̸= NP conjecture [1].\nIn order to circumvent this intrinsic difficulty and possibly to identify\n(or avoid) “hard” instances, random ensembles of optimization problems\nwere introduced and used as test beds for theories and algorithms. This\nline of research has considerably benefited from the methods and concepts\nof statistical mechanics [2,3,4,5]. In particular, a spectacular breakthrough\nwas made by the development of the survey propagation algorithm [5,6]\nwhich is able to solve large random instances of CSPs in the so-called\n“hard-SAT” region. The key to the success of the physics approach lies in\nthe understanding of the rugged energy landscape (reminiscent of glassy\nphases) exhibited by these problems, which survey propagation exploits\nand integrates into a sophisticated message-passing [7] scheme.\nThe structure and organisation of solutions has been analyzed in detail\nfor several CSPs, including the satisfiability problem (k-SAT) [8], the col-\norability of random graphs (k-COL) [9,8,10], and systems of linear Boolean\nequations (k-XORSAT) [11,12,13]. It was shown that as the density of con-\nstraints is increased the space of solutions undergoes several phase transi-\ntions. At low density of constraints the solution space is concentrated in\none big ergodic component, called “cluster” or “state”. For higher densi-\nties the systems undergoes a clustering transition, whereby the solution\nspace breaks into an exponential number of well-separated clusters (this\nseparation can be energetic or entropic). For even higher densities a second\ntransition occurs, which reduces these clusters to a finite number. Finally,\nall clusters disappear at the SAT-UNSAT threshold.\nSince the clustering phenomenon is one of the main building blocks\nunderlying the statistical physics approach1, substantial efforts have been\nmade to give a rigorous base to it [15,16]. Some mathematical results were\nalso obtained in the ergodic phase [17], and in the simple case of linear\nBoolean equations [18]. Remarkably, pure states play an central role in\nspin-glass theory, and they have been extensively studied in that context.\nHowever, the geometrical organization of glassy phases is not yet fully\nunderstood, and the classical picture of complex energy landscapes with\nmany “valleys” still lacks an appropriate representation.\nIn this paper we introduce an exactly solvable random-subcube model2\n(RSM), in the spirit of Derrida’s Random Energy Model (REM) [19]. This\nmodel is inspired by the structure of hard CSPs and optimization prob-\nlems, and reproduces most of their phenomenology. It can be also thought\nof as an attempt to construct a minimal setting that is able to reproduce\nthe structure of solutions in hard CSPs. Its purpose is mainly pedagogi-\n1 Technically, clustering is closely related to the one-step replica-symmetry\nbreaking ansatz used in the replica/cavity method [14,6].\n2 Independently of our work A. Montanari inspired by an idea of D. Achlioptas\nalso introduced this model and worked out some parts of our Sec. 2."},{"page":3,"text":"3\ncal, and it offers an excellent testing playground for ideas and methods in\ncombinatorial optimization and glass physics, while being fully tractable.\nDespite its simplicity, the RSM undergoes the same phase transitions as\nthose observed in random CSPs such as k-SAT [6,8] or k-COL [8,10]. The\nconnection even becomes quantitative in the large-k limit of these problems.\nSo far only the zeroth order of this limit was intuitively known and related\nto Shannon’s random code model [20,21,22], in which clusters are uniformly\ndistributed singletons. One of our most notable results is that the RSM\nprovides the first-order approximation of this large-k limit, reproducing\nthe cluster size distribution and freezing properties of the original models.\nWe also generalize the RSM to deal with continuous energy landscapes\nresembling those observed in glassy systems and hard optimization prob-\nlems, and show how static and dynamical properties can be related explic-\nitly. This energetic RSM displays temperature chaos, undergoes a dynam-\nical transition, and has a Kauzmann temperature.\nThe paper is organized as follows: In Sec. 2 we define the model and\ndescribe its basic properties, as well as the connection with the random k-\nSAT and k-COL problems. We also analyze the behaviour of physics-guided\ndecimation schemes. In Sec. 3 we discuss the relation between its dynamical\nand geometrical properties. In Sec. 4 we extend the definition to energetic\nlandscapes, compute static and dynamical properties, and comment on\nsome ideas from the physics of glassy systems. Finally we present a general\ndiscussion of our results in Sec. 5.\n2 The random-subcube model\n2.1 Definition\nMost constraint satisfaction problems are defined by a set of constraints\non N variables σ = (σ1, . . . , σN) with a finite alphabet, e.g. {0, 1}. In\ncontrast, the random-subcube model is defined directly by its solution space\nS ⊂{0, 1}N; we define S as the union of ⌊2(1−α)N⌋random clusters (where\n⌊x⌋denotes the integer value of x). A random cluster A being defined as:\nA = {σ | ∀i ∈{1, . . . , N}, σi ∈πA\ni },\n(1)\nwhere πA is a random mapping:\nπA : {1, . . . , N} −→{{0}, {1}, {0, 1}}\n(2)\ni 7−→πA\ni\n(3)\nsuch that for each variable i, πA\ni\n= {0} with probability p/2, {1} with\nprobability p/2, and {0, 1} with probability 1−p. A cluster is thus a random\nsubcube of {0, 1}N. If πA\ni\n= {0} or {1}, variable i is said “frozen” in A;"},{"page":4,"text":"4\notherwise it is said “free” in A. One given configuration σ might belong to\nzero, one or several clusters. A solution belongs to at least one cluster.\nThe parameter α is analogous to the density of constraints in CSPs;\nclearly the SAT-UNSAT transition occurs at αs := 1, where clusters cease\nto exist. The parameter p gives the probability that a variable is frozen,\nand plays a role similar to the clause size k in k-SAT, or to the number\nof colors k in k-coloring, as we will see later. Note that in the special case\np = 1, the RSM is equivalent to the random code model with rate R = 1−α\n[21].\nFrozen variables and the structure of the solution space have been in-\ntroduced to mimic the situation observed in random CSPs. However there\nare important differences between the RSM and models like k-SAT. First,\nin real CSPs the clusters are not necessarily subcubes of {0, 1}N. We stress\nhere that when speaking about clusters in the RSM we have in mind the\nabove definition, whereas in the context of the CSPs the notion of cluster is\nmore general [5,8]. Further, in real CSPs the sets of frozen variables associ-\nated with clusters are correlated by the underlying graph, instead of being\ndistributed uniformly. Moreover, free variables in CSPs do not enjoy the\nsame freedom as in the RSM, as clusters usually do not fill up the whole\nsubcube allowed by the frozen variables. In fact, free variables can be corre-\nlated within each cluster in a highly nontrivial way, and these correlations\nmay even be so strong that they create clusters without the help of frozen\nvariables. Clusters without frozen variables are indeed very important, as\ndiscussed recently in [10,23].\n2.2 The basic structural phase transitions\nWe now describe the static properties of the RSM in the thermodynamic\nlimit N →∞(the two parameters α and p being fixed and independent\nof N). The internal entropy s of a cluster A is defined as\n1\nN log2 |A|, i.e.\nthe fraction of free variables in A. The probability P(s) that a cluster has\ninternal entropy s follows the binomial distribution\nP(s) =\n N\nsN\n \n(1 −p)sNp(1−s)N .\n(4)\nLet N(s) be number of clusters of entropy s. This number follows a binomial\nlaw of parameter P(s) with 2N(1−α) terms. Then the mean and the variance\nof N(s) read:\nEN(s) = 2N(1−α)P(s) ,\nVarN(s) = 2N(1−α)P(s)[1 −P(s)] .\n(5)\nBy Markov’s inequality:\nP [N(s) ≥1] ≤E [N(s)] ,\n(6)"},{"page":5,"text":"5\nand by Chebyshev’s inequality:\nP\n \nN(s)\nEN(s) −1\n > ε\n \n≤\nVarN(s)\n[EN(s)]2ε2 ≤\n1\n2N(1−α)ε2P(s)\nfor all ε > 0,\n(7)\nwe get, with high probability (w.h.p.: with probability going to 1 as N →\n∞):\nlim\nN→∞\n1\nN log2 N(s) =\n \nΣ(s) := 1 −α −D(s ∥1 −p) if Σ(s) ≥0,\n−∞\notherwise,\n(8)\nwhere D(x ∥y) := x log2\nx\ny +(1−x) log2\n1−x\n1−y is the binary Kullback-Leibler\ndivergence. Throughout the paper, the same Markov/Chebyshev argument\nwill apply every time we will have to deal with a number of clusters with\na specific property.\nWe now compute the total entropy stot =\n1\nN log2 |S|. First note a ran-\ndom configuration belongs on average to 2N(1−α)(1 −p\n2)N clusters. There-\nfore, if\nα < αd := log2 (2 −p),\n(9)\nthen with high probability the total entropy is stot = 1.\nNow assume α > αd. The total entropy is given by a saddle-point\nestimation:\nX\nA\n2s(A)N = [1 + o(1)]N\nZ\nΣ(s)≥0\nds 2N[Σ(s)+s],\n(10)\nwhence\nstot = max\ns\n[Σ(s) + s | Σ(s) ≥0].\n(11)\nWe denote by s∗= argmax[. . . ] the fraction of free variables in the clusters\nthat dominate the sum. Note that in this sum solutions belonging to several\nclusters have been counted too many times. This does not affect the validity\nof our estimation, since in every cluster the fraction of solutions belonging\nto more than one cluster is exponentially small as long as α > αd.\nDefine ̃s := 2(1 −p)/(2 −p) such that ∂sΣ( ̃s) = −1. The complexity of\nclusters with entropy ̃s reads:\nΣ( ̃s) =\np\n2 −p + log2(2 −p) −α.\n(12)\n ̃s maximizes Eq. (11) as long as Σ( ̃s) ≥0, that is if\nα ≤αc :=\np\n(2 −p) + log2 (2 −p).\n(13)\nThen the total entropy reads\nstot = 1 −α + log2 (2 −p)\nfor\nα ≤αc.\n(14)"},{"page":6,"text":"6\n-0.05\n0\n0.05\n0.1\n0.15\n0.2\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\ns\nslope −1\nsM\nstot\n-0.01\n0\n0.01\n0.02\n0.03\n0.04\n0.45\n0.5\n0.55\n0.6\n0.65\ns\nslope −1\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.8\n0.85\n0.9\n0.95\n1\n1\n0.8\n0.6\n0.4\n0.2\nΣ∗\nΣtot\nm\nstot = sM\nslope −m\nstot\nαc\nNon-Condensed\nCondensed\ns∗\nΣ∗\ns, Σ\nm\ns∗\ns∗= sM\nΣ(s)\nΣ(s)\nα\nFig. 1 (Color online) Top: graphical construction of the maximum of Σ(s) + s\nby a Legendre transformation. In the top left figure (α < αc), the line of slope\n−1 tangent to the complexity function gives the saddle-point s∗, as well as the\ntotal entropy stot by the intercept on the s axis. In the top right figure (α > αc),\nthe supporting line of slope −1 gets “stuck” at sM, where the derivative is −m >\n−1. Bottom: represented as a function of α: total entropy stot, total complexity\nΣtot = 1 −α, typical entropy s∗, complexity of dominating clusters Σ∗= Σ(s∗),\nand m = −∂sΣ(s∗). The condensation point αc marks the separation between\nthe two regimes illustrated above.\nFor α > αc, the maximum in (11) is realized by the largest possible cluster\nentropy sM, which is given by the largest root of Σ(s). Then stot = s∗=\nsM. In this phase the dominating clusters3 have size eNs∗+∆, where ∆=\nO(1) is asymptotically distributed according to a Poisson point process of\nrate e−m∆, i.e., for d∆≪∆the probability that there is at least one state\nof size between eNs∗+∆and eNs∗+∆+d∆is e−m∆d∆, where m = −∂sΣ(s∗).\nExtreme value study of this process leads to the Poisson-Dirichlet [24,25,26]\ndistribution of weights of clusters. In particular it follows that an arbitrary\nlarge fraction of the solutions can be covered by a finite number of clusters.\nSuch a phase is called condensed.\nIn summary, for a fixed value of the parameter p, and for increasing\nvalues of α, four different phases can be distinguished:\n(a) Liquid phase, α < αd: almost all configurations are solutions.\n3 The “dominating clusters” are the minimal set of clusters covering almost all\nsolutions."},{"page":7,"text":"7\n(b) Clustered phase with many states, αd < α < αc: an exponential number\nof clusters is needed to cover almost all solutions.\n(c) Condensed clustered phase, αc < α < αs = 1: a finite number of the\nbiggest clusters cover almost all solutions.\n(d) Unsatisfiable phase, α > αs: no cluster, hence no solution, exists.\nThe very same series of phase transitions is observed in random k-coloring\nand random k-satisfiability, where α is the density of constraints [8,10].\nThe condensation transition at αc corresponds to the Kauzmann temper-\nature [27] in the theory of glasses; at αc the total entropy stot(α) has a\ndiscontinuity in its second derivative with respect to α, in analogy with the\ndiscontinuity of the specific heat at the Kauzmann temperature.\nKeeping in mind the similarity between the RSM and real CSPs, it can\nbe useful to mention some of the properties that are commonly discussed\nin the statistical physics analysis of these problems (for brevity we omit the\nproves of these statements). Among them is the probability distribution of\nmutual overlaps P(q), where q(σ, σ′) = PN\ni=1[2δ(σi, σ′\ni) −1]/N. Below the\ncondensation transition, α < αc, we have P(q) = δ(q), reflecting the fact\nthat random pairs of solutions are uncorrelated. For α > αc, the overlap\nfunction consists of intra-cluster and inter-cluster overlaps: P(q) = wδ[q −\n(1 −stot)] + (1 −w)δ(q), where w is the sum of squares of weights of all\nthe clusters, it is a non-self-averaging random variable, the distribution of\nwhich can be computed from the Poisson-Dirichlet process [26,28].\nAn equivalent way of characterizing the condensed phase is to consider\nthe k-point correlation function, with k ≥2:\nX\nx1,...,xk\n|P(σ1 = x1, σ2 = x2, . . . , σk = xk) −P(σ1 = x1) . . . P(σk = xk)|.\n(15)\nThis quantity decays to zero as N goes to infinity in the non-condensed\nphase, whereas it remains bounded away from zero in the condensed phase.\n2.3 The large k-limit of random k-SAT and k-COL\nOne of the most interesting properties of the random-energy model [19] is\nits equivalence with the large-p limit of the p-spin glass [28]. In the same\nspirit, although the justification is slightly different, the random-subcube\nmodel is found to be equivalent to random k-SAT, random k-COL, and\npresumably other constraint satisfaction problems in the limit k →∞,\nfor connectivities close to the satisfiability threshold. Let us detail this\nstatement.\nIt was already known that at zeroth order (when k →∞) random k-\nSAT and k-COL behave as a random-code model (random-subcube model\nwith p = 1), in which clusters are uniformly distributed singletons. Recent"},{"page":8,"text":"8\nlarge-k calculations of the cluster size distribution Σ(s) in k-SAT and k-\nCOL [8,10] allow a direct comparison with the RSM.\nThe control parameters of the RSM are rescaled as:\np = 1 −ε ,\nα = 1 + ε1 + γ\nln 2 ,\n(16)\nwith ε ≪1 and γ = Θ(1). The cluster size distribution in the RSM is then,\nat leading order:\nΣ(s) ln(2) = s\nh\n1 −ln s\nε\ni\n−ε(2 + γ) + o(ε).\n(17)\nwhile the condensation and satisfiability thresholds read, in terms of the\nrescaled variable γ:\nγc = −2 ln 2 ,\nγs = −1.\n(18)\nIn k-SAT and k-COL, identifying ε and γ in the following way:\nSAT :\nε =\n1\n2k+1 ,\nM\nN = 2k ln 2 −ln 2\n2\n+ γ\n2 ,\n(19)\nCOL :\nε = 1\n2k ,\nM\nN = k ln k −ln k\n2\n+ γ\n2 ,\n(20)\nwhere M is the number of constraints (edges in coloring), and N the number\nof variables, gives a perfect match4 for the complexity function (17), as\ncomputed in [8,10]. This illustrates the analogy between α and the density\nof constraints M/N, as well as between p and k.\nThis equivalence goes further than simply having identical cluster size\ndistributions. The cavity analysis of k-SAT shows that the fraction of free\nvariables in a cluster scales exactly as its internal entropy s. The entropy\nof clusters is thus maximal, from which we infer that clusters fill up the\nwhole subcube prescribed by their frozen variables, like in the RSM. The\nsame is true for k-COL (compare Eqs. (E14) and (E27) in [10]) with the\nsmall difference that for every unfrozen variable only two (out of k) colors\nare allowed.\nNote that this comparison is valid only in a finite vicinity of the satis-\nfiability threshold, for γ = Θk(1). In particular, it does not encompass the\nclustering transition, which for k-SAT (resp. k-COL) occurs for constraint\ndensities scaling as 2k ln 2/k (resp. k ln k/2).\n4 Note the difference in the logarithmic base between here and [8,10]."},{"page":9,"text":"9\n2.4 Decimation\nAn important contribution of statistical physics to the field of combinatorial\noptimization has been to exploit the information provided by message-\npassing algorithms to devise physics-guided decimation schemes.\nMessage-passing algorithms exchange information between units (vari-\nables and constraints) in order to obtain estimates of marginal probabilities\n(beliefs) or other related quantities (e.g. surveys, see below). Subsequently,\nthis information is used to find a solution. A usual way to do this, called\ndecimation, proceeds as follows: fix randomly the value of one5 variable ac-\ncording to its estimated belief, then re-run the message-passing algorithm\non the reduced system, and loop. A trivial statement is that a perfect\nestimate of all marginal probabilities would always cause the decimation\nprocedure to find a solution (if any).\nIn the RSM there is no underlying graph, therefore message-passing can-\nnot be defined. However, it is possible to study decimation schemes based\non exact marginal probability estimators. Although such procedures should\nreally be viewed as thought experiments, they can be used to gain some\ninsight on real algorithms. Here two idealized algorithms are considered:\n– Belief estimator: outputs the exact marginal probabilities μi(σi) =\nP\nσ\\i μ(σ), where μ(σ) = I(σ ∈S)/|S|.\n– Survey estimator: outputs “surveys”, i.e. marginal probabilities over the\nclusters: νi(πi) = P\nπ\\i ν(π), where ν(π) = P\nA I(π = πA)/⌊2N(1−α)⌋.\nIn real CSPs, belief and survey propagation arguably provide asymp-\ntotically accurate estimators, as long as the number of clusters dominating\nthe measure μ (or ν for survey propagation) scales exponentially with N,\nand as long as the one-step replica symmetry breaking description is the\ncorrect one [14,10]. However, belief-guided and survey-guided decimation\nschemes are difficult to analyze in real CSPs — see [29] for an empirical\nstudy on surveys and [30] for recent analytical study on beliefs in k-SAT.\nLet us study decimation in the RSM. As long as the phase is non-\ncondensed, the belief estimator always outputs μi(σi) ≈1/2 for all i in\nthe limit N →∞. Likewise, the survey estimator will output νi({0}) ≈\nνi({1}) ≈p/2, νi({0, 1}) ≈1 −p. In both cases, the decimation procedure\nis completely unbiased: it will fix a random variable i to 0 or 1, with proba-\nbility 1/2. This observation remains true in the subsequent decimation steps\nas long as the number of clusters dominating the reduced measure μ (or\nν) remains exponential. Within this assumption, after T = tN (0 ≤t ≤1)\ndecimation steps, T variables will be fixed randomly and independently.\nWe are then left with a restricted space of solutions compatible with these\nT fixed variables. The logarithm of the number of clusters of entropy sN\n5 In practice the number of variables fixed at each step can range from one to\na small fraction of the variables."},{"page":10,"text":"10\nis then\nNΣt(s) = N\n \n1 −α + t log2\n \n1 −p\n2\n \n−(1 −t)D\n \ns\n1 −t||1 −p\n \n.\n(21)\nRescaling by the number of unfixed variables (1 −t)N: Σt = (1 −t) ̄Σt,\ns = (1 −t) ̄s, we obtain\n ̄Σt( ̄s) = 1 −α −tαd\n1 −t\n−D( ̄s||1 −p).\n(22)\nThe parameter ̄α(t) := (α −tαd)/(1 −t) now plays the same role as α\nin the analysis of the RSM. Consequently, the system undergoes the same\ncondensation and unsatisfiability transitions as t is increased. Assume for\nexample that αd < α < αc. Fixing a fraction tc := (αc −α)/(αc −αd)\nof the variables will cause the system to condense. The belief estimator\nwill then be dominated by a finite number of clusters, yielding instance-\ndependent biases on variables. An extensive number of variables become\nsuddenly near-frozen, i.e., μi(1) ≈0 or ≈1, and remain so for t > tc. At\nts := (1 −α)/(1 −αd), the total complexity goes to zero, and near-frozen\nvariables become truly frozen, i.e. μi(1) = 0 or 1, as all sub-dominant\nclusters disappear. By contrast, the survey estimator will output unbiased\nmarginal probabilities as long as the total number of clusters is exponential\nin N, that is if t < ts. At t = ts, decimation concentrates on a single\ncluster, causing a freezing avalanche in the surveys, i.e. for each variable\ni, νi({0, 1}) = 1 or νi({0}) = 1 or νi({1}) = 1. For both estimators, any\nslight error at ts will cause the failure of the decimation process.\nNote that in real CSPs, belief propagation is not expected to be correct\nin the condensed phase, beyond tc; in fact, it will not detect condensation,\nnor near-frozen variables. For a recent study of the belief-propagation-\nguided decimation in k-SAT see [30]. Remarkably, in real CSPs, survey-\nguided decimation usually simplifies the problem: after a certain number\nof decimation steps, it outputs νi({0, 1}) = 1 for all i, and the problem can\nthen be passed over to a simple local search algorithm [6,29]. Although the\nRSM, which is intrinsically clustered, is unable to capture this property, it\nsheds some light on why decimation may not work in some cases.\n3 Distance and ergodicity\nThe geometrical organization of solutions to CSPs is thought to play an\nimportant role in setting intrinsic limits to the performance of search al-\ngorithms. In particular, the clustering phenomenon, by which the solution\nspace is fragmented into many connected components far from each other,\nhas been proposed by physicists as a possible explanation for the failure of\nmost known algorithms [4,5,6]. Also, the role of frozen variables was re-\ncently discussed in [10]. Conversely, the success of survey propagation [6] is"},{"page":11,"text":"11\nusually explained by the fact that it explicitly incorporates the existence of\nclusters. The separability of clusters has been proved in the k-SAT problem,\nin compliance with the predictions of statistical physics [15]. Despite this\nevidence, the precise relation between geometry and algorithms still lacks a\nrigorous base. The random-subcube model offers an excellent opportunity\nto study these questions in a well controlled framework.\n3.1 The dynamical transition\nLet us first argue why the clustering transition at αd, defined by Eq. (9),\nactually corresponds to what is commonly refered to as the dynamical\n(ergodicity breaking) transition in real CSPs [31,32,33,8,12,10]. To this\nend we study a uniform unbiased random walk on the space of solutions:\nat each step, one is allowed to move from one solution to the other by\nflipping only one spin. We choose this dynamics for the sake of simplicity,\nbut most of the arguments below hold for more general dynamical rules,\nlike for example the flipping of a sub-extensive number of spins at each\nstep.\nWe have already pointed out that an arbitrary configuration belongs\nto 2(1−α)N(1 −p/2)N different clusters w.h.p. if α < αd, and to none if\nα > αd. Therefore below αd almost all configurations are solutions, and\nany reasonable dynamics will explore the entire phase space uniformly.\nOn the other hand, when α > αd, solutions become exponentially rare.\nLet A be a cluster of internal entropy s. What is the probability that after t\nsteps, a random walker ends up in another cluster B of internal entropy s′?\nLet a denote the proportion of variables that are free in A and frozen in B.\nThe probability distribution of a is given by the total number of partitions\nof {1, . . . , N} into four categories: frozen in A and B, only frozen in A, only\nfrozen in B, and frozen neither in A nor B. This probability reads:\nq(a) =\n1\n N\nNs\n N\nNs′\n \nN!\n(Na)![N(s −a)]![N(1 −s′ −a)]![N(s′ −s + a)]!.\n(23)\nIn order for the walker to reach B from A, it has to match perfectly the\nprescriptions (freezings) of B on these aN variables. If t = Θ(N d), the prob-\nability of this happening is ≈t2−aN. Additionally, variables that are frozen\nin both clusters must coincide, so that A and B have a non-empty intersec-\ntion. For a random choice of B, this happens with probability 2N(s′+a−1).\nConsequently, the probability that the walker wanders in any other\ncluster of entropy s′ after t steps is union-bounded by:\nφ(s →s′) ≤2NΣ(s′) X\na\nq(a) t 2−aN 2N(s′+a−1)\n(24)"},{"page":12,"text":"12\nThe maximum of\n1\nN log q(a) tends to zero when N →∞, so that:\nlim sup\nN→∞\n1\nN log φ(s →s′) ≤Σ(s′) + s′ −1\n(25)\nThis quantity remains negative for all s′, as long as α > αd. Therefore,\nin this regime, hopping from one cluster to the other is very unlikely in\na sub-exponential number of steps, even though clusters are not disjoint.\nIn this sense we say that ergodicity is broken. Remarkably, the space of\nsolutions becomes non-ergodic as soon as it becomes non-trivial, at αd.\nLet us stress the importance of the entropic barriers between clusters\nin our analysis. In the real CSPs and optimization problems the energetic\nbarriers are usually thought of as more important, and clusters are even\nsometimes described as separated (by an extensive distance). The impor-\ntance of entropic barriers and the possibility of non-extensively separated\nclusters should, however, not be neglected in the studies of richer models.\n3.2 x-satisfiability\nThe notion of x-satisfiability was first introduced as a tool to study the\ngeometrical structure of the solution space of CSPs [15]. An instance of CSP\nis said x-satisfiable if and only if it admits a pair of solutions separated by a\nHamming distance ∼xN. In other words, x-satisfiability gives the distance\nspectrum of the solution space. This spectrum is estimated using three\nquantities:\na) d1 = x1(α)N: the maximum distance between two solutions inside one\ncluster,\nb) d2 = x2(α)N: the minimum distance between two solutions from two\ndistinct clusters,\nc) d3 = x3(α)N: the maximum distance between any two solutions (pre-\nsumably from two different clusters).\nThe first of these quantities is estimated by noting that the maximum\ndistance between any two solutions in a given cluster, i.e. its diameter,\nequals its entropy s. Therefore the maximum diameter/entropy x1 is given\nw.h.p. by the largest internal entropy sM, i.e. the largest root of Σ(s) =\n1 −α −D(s ∥1 −p).\nNow take two clusters A and B at random, and consider the probability\nthat their distance be xN. This distance is given by the number of variables\nwhich are frozen in both clusters, but in a contradictory way, such that\nπA(i) ̸= πB(i). This happens independently with probability p2/2 for each\nvariable, so that the number of such variables follows a binomial law of\nparameter p2/2. Therefore, the number N(x) of pairs of clusters at distance"},{"page":13,"text":"13\nαs(x)\nx1\nx2\nαgap\nαsep\nαc\nx\nα\n1\n0.9\n0.8\n0.7\n0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0\n1\n0.9\n0.8\n0.7\n0.6\nFig. 2 (Color online) The x-satisfiability threshold is constructed from the three\ndistances x1(α), x2(α) and x3(α). Below a threshold αgap, distance spectra fail to\ndetect the fragmentation of the solution space (there is no more “gap” between\nintra and inter-cluster distances). Below another threshold αsep, clusters cease\nto be all well separated, although ergodicity is still broken. The condensation\nthreshold αc is shown for information. The dynamical threshold αd lies outside\nthe picture, and its value is ≈0.070. In this figure p = 0.95.\nxN coincides w.h.p. with its mean value:\nE[N(x)] = 22(1−α)N\n N\nNx\n \n1 −p2\n2\n (1−x)N p2\n2\n xN\n≍2Ns2(x)\n(26)\nif s2(x) := 2(1 −α) −D(x ∥p2/2) > 0, and N(x) = 0 w.h.p. if s2(x) < 0.\nConsequently the smallest possible distance between any two clusters is\ngiven by x2N, where x2(α) is the smallest root of s2(x). A similar argument\ngives the largest distance between any two solutions from two different\nclusters: x3(α) = 1 −x2(α).\nTo sum up, we find that a random instance is x-satisfiable w.h.p. if\nα < αs(x), and is x-unsatisfiable w.h.p. if α > αs(x), with:\nαs(x) =\n \n \n \n \n \n \n \n1\nif x ∈[0, 1 −p] ∪[p2/2, 1 −p2/2]\n1 −D(x ∥1 −p)\nif x ∈[1 −p, x0]\n1 −1\n2D(x ∥p2/2)\nif x ∈[x0, p2/2]\n1 −1\n2D(1 −x ∥p2/2) if x ∈[1 −p2/2, 1]\n(27)\nwhere x0 is solution to D(x ∥p2/2) = 2D(x ∥1 −p)."},{"page":14,"text":"14\nFig. 2 shows how αs(x) can be constructed from the three distances x1,\nx2, x3. We put αsep := 1 + (1/2) log(1 −p2/2) > αd, the threshold below\nwhich some pairs of clusters have a non-empty intersection. An interesting\nobservation is that ergodicity can still be broken even below this threshold.\nWe can also define the αgap := αs(x0) > αsep, below which distances\nfrom the same cluster and distances from distinct clusters overlap. This\nthreshold sets the limit below which the notion of x-satisfiability fails to\ndetect clustering. The random-subcube model allows us to make a clear\nand intelligible distinction between the three thresholds αd < αsep < αgap.\nWe expect this distinction to hold in most CSPs6.\n4 Random energy landscape\n4.1 Definition\nThe random-subcube model can be enriched by adding the notion of energy\nto the definition of states. The motivation for doing this is to mimic the\noptimization version of CSPs (where energy is defined as the number of un-\nsatisfied constraints), but it can also be used to reproduce some properties\nof glassy systems.\nFor each energy level E0, we define N(E0) = ⌊2NΣ(E0/N)⌋valleys of\nenergy E0, where Σ(e0) is an increasing complexity function. This function\ncan be arbitrary, but for simplicity we will restrict our examples to the\nform7 Σ(e0) = a + be0 −ce0 ln(e0), where a > 0 corresponds to a SAT\nphase, and a < 0 to an UNSAT phase.\nEach valley is defined as a subcube V , chosen at random in the same\nway as clusters in the previous sections, cf. (1). In the following, the freezing\nprobability p will be fixed for all energies, but one could easily generalize\nthe model by making it energy-dependent: p = p(e0).\nThe number N(E0, S0) of valleys of energy E0 = e0N and entropy\nS0 = s0N is w.h.p.:\nN(E0, S0) ≍2NΣ(e0,s0) if Σ(e0, s0) := Σ(e0) −D(s0 ∥1 −p) ≥0\n= 0\notherwise.\n(28)\nGiven a configuration σ, we define its energy as a trade-offbetween the\nenergy of surrounding valleys and their distance. Let us denote the energy\nof a valley V by E0(V ). Then the energy of σ is:\nE(σ) := min\nV\n[E0(V ) + d(σ, V )]\n(29)\n6 With the notable exception of k-XORSAT, where αd = αsep. Incidentally in\nk-XORSAT we also have αc = αs.\n7 This form corresponds to the function used to fit data from the cavity method\nin the k-SAT problem [6]."},{"page":15,"text":"15\nwhere d(σ, V ) is the distance between σ and the nearest element of V .\nBy definition, we say that σ belongs to the basin of attraction of V if\nV minimizes the sum. Observe that with this definition, it may happen\nthat some valleys are not represented at all in the energy landscape. In the\nfollowing, the term “state-bottom energy” shall refer to the energy E0 of\nthe valley minimizing the sum, and the term “state-bottom entropy” to the\nentropy S0 of that valley.\n4.2 Static description of the energy landscape\nWhat is the energy of an arbitrary configuration σ? Let us start with\nthe typical case: for each e0 = E0/N, we compute the distance to the\nnearest valley with energy E0. Standard arguments show that the number\nof valleys of state-bottom energy e0N at distance d = ωN is governed by\nthe exponent:\nΣ(e0) −D(ω ∥p/2).\nThen, the minimum distance is given, in the N\n→\n∞limit, by\nδ[Σ(e0), p/2]N, where δ(x, y) is solution to: x = D(δ ∥y). Then, the typi-\ncal energy is obtained as the best compromise between state-bottom energy\nand distance:\ne∗= E/N = min\ne0 {e0 + δ[Σ(e0), p/2]} ,\n(30)\nThe argmin gives the typical state-bottom energy e∗\n0 of a random σ.\nAs we just saw, most configurations have roughly the same energy, and\nbelong to valleys with the same state-bottom energy. At finite temperature\nhowever, thermodynamics will be dominated by configurations of lower\nenergy than e∗N. We thus need to estimate the entropy function (governing\nthe number of configurations of given energy), the Legendre transform of\nwhich shall give us the free energy. Given a valley V of energy E0 = e0N\nand entropy S0 = s0N, the number of configurations of energy E belonging\nto this valley is:\nNV (E) = 2S0\n E −E0\nN −S0\n \n,\n(31)\n≍2NsV (e|e0,s0),\nwith\nsV (e|e0, s0) := s0 + (1 −s0)H\n e −e0\n1 −s0\n \n,\nwhere H(x) = −x log2 x−(1−x) log2 (1 −x) is the entropy function. Sum-\nming up over all valleys, the total number of configurations with energy\nE = eN is:\nN(E) =\nX\nS0,E0\n2S0\n E −E0\nN −S0\n \n2NΣ(E0/N,S0/N) ≍2Ns(e),\n(32)\nwith\ns(e) =\nmax\ne0,s0\nΣ(e0,s0)≥0\n[sV (e|e0, s0) + Σ(e0, s0)] .\n(33)"},{"page":16,"text":"16\nHere we have implicitly assumed that all elements in the sphere of radius\nE−E0 and center V are in the basin of attraction of V , as long as E < e∗N.\nThis is not true in general, as some configurations may in fact belong to\na more favorable basin, and may thus have lower energies. However, such\nconfigurations remain exponentially rare in comparison to the total weight\nof the sphere. Therefore the previous estimate holds8.\nCanceling the derivative w.r.t. s0 in (33) yields the saddle for s0:\n ̃s0 = (1 −p)(1 −e + e0)\n1 −p/2\n(34)\nProvided that the maximum is reached in a region where Σ(e0, ̃s0) > 0, we\nget:\ns(e) = max\ne0 [1 −D(e −e0 ∥p/2) + Σ(e0)]\nfor\nec < e < e∗.\n(35)\nThis will be valid from e = e∗(for which we find s(e∗) = 1 as expected)\ndown to a certain condensation energy ec. Below that energy (e ≤ec), the\nphase is condensed: Σ(e0, ̃s0) < 0, and the maximum in (33) is reached on\nthe border on the definition domain, where Σ(e0, s0) = 0. If we denote by\nsM(e0) the biggest valley of energy e0 (i.e. the largest of root of Σ(e0, s0) =\n0), we get:\ns(e) = max\ne0 {sV [e|e0, sM(e0)]}\nfor\ne < ec.\n(36)\nSo far we have worked in the microcanonical ensemble, but the same ar-\nguments hold in the canonical ensemble. In particular the condensation\ntemperature is Tc = (∂s/∂e|e=ec)−1. The number of dominating states in\nthe condensed phase follows again a Poisson-Dirichlet process [25,26] with\nparameter m, where m/T is the slope of the curve ΣT (f) at its smallest\nroot. The function ΣT (f) is the canonical counterpart of Σ(e0, s0), and\nreads:\nΣT (f) =\nmax\ne0,s0:fV (T |e0,s0)=f Σ(e0, s0) ,\n(37)\nwhere fV (T |e0, s0) is the single-state free energy, obtained as the Legendre\ntransform of sV (e|e0, s0) in (31).\nIn both low-temperature phases (condensed and non-condensed), equi-\nlibrium is reached for different values of (e0, s0) as e varies. Said differently,\nthe states dominating the microcanonical measure at e and at e + δe are\ncompletely distinct. In the canonical language, we say that the system ex-\nhibits temperature chaos [34,35]: slightly changing the temperature from T\nto T + δT dramatically modifies the free-energy landscape, reshuffling the\nordering of states. Consequently, correlations are nonexistent between T\n8 This argument is similar to the one used in Eq. (10), where some solutions\nwere counted several times, but with no consequence at the exponential scale."},{"page":17,"text":"17\n 0.8\n 0.82\n 0.84\n 0.86\n 0.88\n 0.9\n 0.92\n 0.94\n 0.96\n 0.98\n 1\n 0.03\n 0.04\n 0.05\n 0.06\n 0.07\n 0.08\nV3\nV2\nV1\nsV2(e)\nsV3(e)\nslope 1/T\nsV1(e)\nsM(e0)\ns\ne\nFig. 3 (Color online) Illustration of the temperature chaos by construction of the\nmicrocanonical entropy in the condensed phase, cf. Eq. (36), and level crossing.\nWe have represented three entropy curves sV (e) corresponding to three extremal\nstates V1, V2 and V3, the envelope of all these curves is the microcanonical entropy.\nThese states are maximally atypical; they realize a balance between low state-\nbottom energies and a high state-bottom entropies, which are related by s0 =\nsM(e0) (thick curve). As the temperature (or energy) is decreased, the curves\nsV (e) cross each other, and the system is dominated by states of lower state-\nbottom energies and entropies. These data were obtained for number of valleys\nΣ(e0) = −0.05 −0.5e0 ln e0 and p = 0.2.\nand T +δT . This phenomenon of free-energy crossings is illustrated (in the\ncondensed phase) by Fig. 3, where the maximum of Eq. (36) is constructed\ngeometrically.\n4.3 Relation with dynamics\nWe now undertake to describe the dynamical properties of this energetic\nlandscape. To that end we shall make use of the static picture, which we\nknow precisely from bottom-up construction.\nOur reasoning proceeds in two steps. First, we study the behavior of\na single spin-flip Monte-Carlo dynamics with detailed balance evolving in\na single valley of state-bottom energy e∗\n0, and state-bottom entropy s∗\n0. In\na second step, we argue that the same dynamics run on the full rugged\nenergy landscape is entirely governed by this single-state-like behavior.\nThe thermodynamics of a single typical state is given by:\ns(e|e∗\n0, s∗\n0) = s∗\n0 + (1 −s∗\n0)H\n e −e∗\n0\n1 −s∗\n0\n \n(38)"},{"page":18,"text":"18\nwhere s∗\n0 = (1−p)(1−e∗+e∗\n0)/(1−p/2) is the typical state-bottom entropy\nof a random configuration, computed from Eq. (34). Supposing now that the\nenergy landscape is made of this state only, we can easily convince ourselves\nthat the dynamics is ergodic: the energy landscape is convex, and there are\nneither energetic nor entropic barriers. Both the quenched and annealed9\ndynamics in this simple landscape is then given by the thermodynamics.\nWe know from the previous paragraphs that states extend up to energy\ne∗, where a “crest” connects the different valleys. This crest is actually more\nof a plateau, as it embeds almost all configurations: s(e∗) = 1. Therefore,\nour dynamics will remain there as long as the temperature does not al-\nlow configurations of lower energy. This happens at Td := (∂s/∂e|e=e∗)−1,\nwhere exploring valleys starts to be more favorable.\nBelow Td, the system will find itself trapped in one state, since barri-\ners between valleys are extensive. This randomly picked state has typical\nproperties: in particular, its bottom-energy is e∗\n0 and its bottom-entropy is\ns∗\n0.\nWe argue that below Td everything happens as if this trapping state was\nput in isolation, as we have described above. The justification comes from\nthe fact that a valley does not “see” its neighbors as long as e < e∗: even\nthough many configurations of energy e < e∗in the isolated state actually\nbelong other valleys in the full landscape, their proportion is exponentially\nsmall. In other words, although there may be some directions for which the\nenergy barrier is lower than e∗−e, these directions are beaten entropically.\nWith this reasoning, both the quenched and annealed dynamics are\ndescribed by the microcanonical entropy of a single typical state: sdyn(e) =\ns(e|e∗\n0, s∗\n0). We note in passing:\nT −1\nd\n= ∂s\n∂e\n \ne=e∗= ∂sdyn\n∂e\n \ne=e∗,\n(39)\nas partial derivatives w.r.t. e0 and s inside (33) cancel at the maximum.\nThe dynamical temperature Td is thus well defined.\nNote that this analysis is exact: the fact that it purely relies on static\narguments should be attributed to the simplicity of the model. In richer\nmean-field models this kind of arguments might also apply, but will most\nprobably not give the full picture.\n4.4 Glassy behaviour\nThe analysis of the energy landscape in the RSM is summarized in the\nenergy-temperature phase diagram of Fig. 4. As anticipated, the behaviour\n9 By quench, resp. annealing, we mean a fast, resp. slow, change of temperature\nfrom T1 > Td to T2 < Td and some time (a finite number of Monte Carlo sweeps)\nspent at T2 afterwards."},{"page":19,"text":"19\n0\n0.5\n1\nm\n0\n0.2\n0.4\n0.6\n0.8\n1\n1.2\n1.4\nT\n0\n0.2\n0.4\n0.6\n0.8\n1\n1.2\n1.4\nT\n0\n0.05\n0.1\n0.15\n0.2\n0.25\nenergy\ne∗\nec\negs\ne∗\n0\nTd\nTc\ncondensed glass\nglass\nliquid\ndynamical\nFig. 4 (Color online) Energy as a function of temperature. At temperature\nT > Td, the system is in a liquid state: the dynamics is exploring ergodically\nall configurations of energy e∗. Below the dynamical temperature ergodicity is\nbroken. The upper “dynamical” curve shows the result of a quench/annealing\nin temperature, whereby the systems remains stuck in a typical state of bot-\ntom energy e∗\n0. For Tc < T < Td, equilibrium thermodynamics is dominated\nby an exponential number of states (curve “glass”). Below the condensation\n(Kauzmann) temperature thermodynamics is dominated by a finite number of\nstates (curve “condensed glass”). This number is given by a Poisson-Dirichlet\nprocess of parameter m (plotted in the upper part of the diagram). The dashed\nline shows the result of a quench/annealing starting from an equilibrium state\nat temperature Tc < T < Td. The bottom line shows the bottom energies\nof the thermodynamically dominating states. These curves were obtained for\nΣ(e0) = −0.05 −0.5e0 ln e0 and p = 0.6.\nof the RSM resembles the one observed in glasses and spin-glasses. The\ntwo distinct glassy transitions (dynamical and condensation), as well as\nthe phenomenon whereby the physical dynamics gets stuck in metastable\nstates, have been described for example in the p-spin glass [28], the spherical\np-spin glass [36], the Potts glass [37] and the lattice glass [38]. Several\nrelated examples of energy-temperature diagrams were derived recently in\n[39].\nIn the aforementioned mean-field models, the static behaviour is bet-\nter understood than the dynamics. Static properties are usually analyzed\nby the replica/cavity method, with the help of Parisi’s replica symmetry\nbreaking scheme. A satisfactory analytic treatment of the dynamics exists"},{"page":20,"text":"20\nonly for the spherical p-spin glass [36], in which all states have the same\nentropy. A remarkable step towards connecting the static picture and dy-\nnamical behaviour in a rather general framework was done in [31,32,33].\nThe RSM could provide a tractable playground for studying several aspects\nof glassy dynamics, e.g. aging and rejuvenation [40].\nWe would like to emphasize the freedom we can enjoy in the definition\nof the energy landscape. First, arbitrary numbers and sizes of valleys at\neach energy e0, Σ(e0) and p(e0), can be considered. Second, the definition\nof the configurational energy E(σ) in Eq. (29) could be generalized to an\narbitrary function of all valleys V and σ; E(σ) = F({V }, σ). By tun-\ning these parameters, one could hope to reproduce the dynamics of more\ncomplex models on a quantitative level.\n5 Conclusions\nThe random-subcube model is a simple exactly solvable model capturing\nseveral interesting properties of random constraint satisfaction problems.\nRather than an attempt to construct a new realistic model for practical\ninstances of constraint satisfaction problems, it allows us to identify which\nproperties of random CSPs can be reproduced by a simple probabilistic\nstructure, and conversely, which of these properties may be intrinsically\nnon-trivial. Examples of reproducible properties include condensation, non-\nmonotony of the x-satisfiability threshold, temperature chaos and dynami-\ncal freezing in metastable states. From this point of view, the RSM stands\njust next to the random-energy model [19], the random-code model [20,21,\n22] or the random-energy random-entropy model [35].\nSince the relation between the RSM and the large-k limit of random\nk-SAT and k-COL is based on non-rigorous results from [8,10], it would\nbe interesting to establish this equivalence rigorously. Further, the RSM\nshould be helpful for understanding some properties which are too difficult\nto study in more realistic models, such as finite-size corrections or some\naspects of glassy dynamics.\nFinally, our work addresses the broad question of producing, infer-\nring, and representing complex and rugged structures of the hypercube.\nAlthough we retained the simplest choice of subcubes for clusters, more so-\nphisticated alternatives could be explored and used to reproduce detailed\ngeometrical features of solution spaces in CSPs.\n6 Acknowledgment\nWe would like to thank Florent Krz ֒aka la and Marc M ́ezard for fruitful\ndiscussions about this work, and Guilhem Semerjian and Jiˇr ́ı ˇCern ́y for\ncritical reading of the manuscript. This work has been partially supported\nby EVERGROW (EU consortium FP6 IST)."},{"page":21,"text":"21\nReferences\n1. C. H. Papadimitriou. Computational complexity. Addison-Wesley, 1994.\n2. S. Kirkpatrick and B. Selman. Critical behavior in the satisfiability of random\nboolean expression. Science, 264:1297–1301, 1994.\n3. R. Monasson, R. Zecchina, S. Kirkpatrick, B. Selman, and L. Troyansky.\nDetermining computational complexity from characteristic phase transitions.\nNature, 400:133–137, 1999.\n4. G. Biroli, R. Monasson, and M. Weigt.\nA variational description of the\nground state structure in random satisfiability problems. Eur. Phys. J. B,\n14:551, 2000.\n5. M. M ́ezard, G. Parisi, and R. Zecchina. Analytic and algorithmic solution of\nrandom satisfiability problems. Science, 297:812–815, 2002.\n6. M. M ́ezard and R. Zecchina.\nRandom k-satisfiability problem: From an\nanalytic solution to an efficient algorithm. Phys. Rev. E, 66:056126, 2002.\n7. F. R. Kschischang, B. Frey, and H.-A. Loeliger. Factor graphs and the sum-\nproduct algorithm. IEEE Trans. Inform. Theory, 47(2):498–519, 2001.\n8. Florent Krzakala, Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Se-\nmerjian, and Lenka Zdeborov ́a.\nGibbs states and the set of solutions of\nrandom constraint satisfaction problems. Proc. Natl. Acad. Sci., 104:10318,\n2007.\n9. M. M ́ezard, M. Palassini, and O. Rivoire. Landscape of solutions in constraint\nsatisfaction problems. Phys. Rev. Lett., 95:200202, 2005.\n10. L. Zdeborov ́a and F. Krzakala. Phase transitions in the coloring of random\ngraphs. Phys. Rev. E, 76:031131, 2007.\n11. S. Cocco, O. Dubois, J. Mandler, and R. Monasson. Rigorous decimation-\nbased construction of ground pure states for spin glass models on random\nlattices. Phys. Rev. Lett., 90:047205, 2003.\n12. M. M ́ezard, F. Ricci-Tersenghi, and R. Zecchina. Alternative solutions to\ndiluted p-spin models and XORSAT problems. J. Stat. Phys., 111:505, 2003.\n13. T. Mora and M. M ́ezard. Geometrical organization of solutions to random\nlinear Boolean equations. Journal of Statistical Mechanics: Theory and Ex-\nperiment, 10:P10007, October 2006.\n14. M. M ́ezard and G. Parisi. The bethe lattice spin glass revisited. Eur. Phys.\nJ. B, 20:217, 2001.\n15. M. M ́ezard, T. Mora, and R. Zecchina. Clustering of solutions in the random\nsatisfiability problem. Physical Review Letters, 94:197205, 2005.\n16. Dimitris Achlioptas and Federico Ricci-Tersenghi. On the solution-space ge-\nometry of random constraint satisfaction problems. In STOC ’06: Proceedings\nof the thirty-eighth annual ACM symposium on Theory of computing, pages\n130–139, New York, NY, USA, 2006. ACM Press.\n17. A. Montanari and D. Shah. Counting good truth assignments of random k-\nsat formulae. In Proceedings of the 18th Annual ACM-SIAM Symposium on\nDiscrete Algorithms, pages 1255–1264, New York, USA, 2007. ACM Press.\n18. O. Dubois and J. Mandler. The 3-xorsat threshold. FOCS, 00:769, 2002.\n19. B. Derrida. Random-energy model: Limit of a family of disordered models.\nPhys. Rev. Lett, 45:79–82, 1980.\n20. C. E. Shannon. A mathematical theory of communication. Bell System Tech.\nJournal, 27:379–423, 623–655, 1948.\n21. A. Montanari. The glassy phase of Gallager codes. Eur. Phys. J. B., 23:121–\n136, 2001.\n22. A. Barg and G. D. Forney Jr. Random codes : minimum distances and error\nexponents. IEEE Trans. Inform. Theory, 48:2568–2573, 2002.\n23. Guilhem Semerjian. On the freezing of variables in random constraint sat-\nisfaction problems.\nJ. Stat. Phys., 130:251, 2008.\n(to appear) preprint\narXiv.org:0705.2147."},{"page":22,"text":"22\n24. M. M ́ezard, G. Parisi, N. Sourlas, G. Toulouse, and M. A. Virasoro. Replica\nsymmetry breaking and the nature of the spin-glass phase.\nJ. Physique,\n45:843–854, 1984.\n25. M. Talagrand. Rigorous low temperature results for the p-spin mean field\nspin glass model. Probability Theory and Related Fields, 117:303–360, 2000.\n26. J. Pitman and M. Yor.\nThe two-parameter poisson-dirichlet distribution\nderived from a stable subordinator. Ann. Probab., 25:855–900, 1997.\n27. W. Kauzmann. The nature of the glassy state and the behavior of liquids at\nlow temperatures. Chem. Rev., 43:219, 1948.\n28. D.J. Gross and M. M ́ezard. The simplest spin glass. Nucl. Phys. B, 240:431,\n1984.\n29. G. Parisi.\nSome remarks on the survey decimation algorithm for k-\nsatisfiability. arXiv:cs/0301015, 2003.\n30. A. Montanari, F. Ricci-Tersenghi, and G. Semerjian.\nSolving con-\nstraint satisfaction problems through belief propagation-guided decimation.\narXiv:0709.1667v1 [cs.AI], 2007.\n31. A. Montanari and G. Semerjian. From large scale rearrangements to mode\ncoupling phenomenology. Phys. Rev. Lett., 94:247201, 2005.\n32. A. Montanari and G. Semerjian. On the dynamics of the glass transition on\nbethe lattices. J. Stat. Phys., 124:103–189, 2006.\n33. A. Montanari and G. Semerjian. Rigorous inequalities between length and\ntime scales in glassy systems. J. Stat. Phys., 125:23, 2006.\n34. A. J. Bray and M. A. Moore. Chaotic nature of the spin-glass phase. Phys.\nRev. Lett., 58:57–60, 1987.\n35. F. Krzakala and O. C. Martin. Chaotic temperature dependence in a model\nof spin glasses. Eur. Phys. J. B., 28:199–208, 2002.\n36. L. F. Cugliandolo and J. Kurchan. Analytical solution of the off-equilibrium\ndynamics of a long-range spin glass model. Phys. Rev. Lett., 71:173, 1993.\n37. D. J. Gross, I. Kanter, and H. Sompolinsky. Mean-field theory of the potts\nglass. Phys. Rev. Lett., 55(3):304–307, Jul 1985.\n38. G. Biroli and M. M ́ezard. Lattice glass models. Phys. Rev. Lett., 88:025501,\n2002.\n39. F. Krzakala and J. Kurchan. A landscape analysis of constraint satisfaction\nproblems. Phys. Rev. B, 76:021122, 2007.\n40. J.-L. Barrat, M.V. Feigelman, J. Kurchan, and J. Dalibard. Slow Relaxations\nand Nonequilibrium Dynamics in Condensed Matter Les Houches Session\nLXXVII, 1-26 July, 2002. Springer, Berlin, 2003."}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"time, as formalized by the P ̸= NP conjecture [1].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"on N variables σ = (σ1, . . . , σN) with a finite alphabet, e.g. {0, 1}. In","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"A = {σ | ∀i ∈{1, . . . , N}, σi ∈πA","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"= {0} with probability p/2, {1} with","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"= {0} or {1}, variable i is said “frozen” in A;","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"clearly the SAT-UNSAT transition occurs at αs := 1, where clusters cease","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"p = 1, the RSM is equivalent to the random code model with rate R = 1−α","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"P(s) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"EN(s) = 2N(1−α)P(s) ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"VarN(s) = 2N(1−α)P(s)[1 −P(s)] .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"N log2 N(s) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"Σ(s) := 1 −α −D(s ∥1 −p) if Σ(s) ≥0,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"where D(x ∥y) := x log2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"We now compute the total entropy stot =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"α < αd := log2 (2 −p),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"then with high probability the total entropy is stot = 1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"2s(A)N = [1 + o(1)]N","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"stot = max","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"We denote by s∗= argmax[. . . ] the fraction of free variables in the clusters","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"Define ̃s := 2(1 −p)/(2 −p) such that ∂sΣ( ̃s) = −1. The complexity of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"Σ( ̃s) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"α ≤αc :=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"stot = 1 −α + log2 (2 −p)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"stot = sM","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"s∗= sM","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"Σtot = 1 −α, typical entropy s∗, complexity of dominating clusters Σ∗= Σ(s∗),","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"and m = −∂sΣ(s∗). The condensation point αc marks the separation between","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"entropy sM, which is given by the largest root of Σ(s). Then stot = s∗=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"sM. In this phase the dominating clusters3 have size eNs∗+∆, where ∆=","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"of size between eNs∗+∆and eNs∗+∆+d∆is e−m∆d∆, where m = −∂sΣ(s∗).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"(c) Condensed clustered phase, αc < α < αs = 1: a finite number of the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"mutual overlaps P(q), where q(σ, σ′) = PN","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"i=1[2δ(σi, σ′","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"condensation transition, α < αc, we have P(q) = δ(q), reflecting the fact","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"function consists of intra-cluster and inter-cluster overlaps: P(q) = wδ[q −","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"|P(σ1 = x1, σ2 = x2, . . . , σk = xk) −P(σ1 = x1) . . . P(σk = xk)|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"with p = 1), in which clusters are uniformly distributed singletons. Recent","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"p = 1 −ε ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"with ε ≪1 and γ = Θ(1). The cluster size distribution in the RSM is then,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"Σ(s) ln(2) = s","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"γc = −2 ln 2 ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"γs = −1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"N = 2k ln 2 −ln 2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"N = k ln k −ln k","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"fiability threshold, for γ = Θk(1). In particular, it does not encompass the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"– Belief estimator: outputs the exact marginal probabilities μi(σi) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"σ\\i μ(σ), where μ(σ) = I(σ ∈S)/|S|.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"clusters: νi(πi) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"π\\i ν(π), where ν(π) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"A I(π = πA)/⌊2N(1−α)⌋.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"ν) remains exponential. Within this assumption, after T = tN (0 ≤t ≤1)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"NΣt(s) = N","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"Rescaling by the number of unfixed variables (1 −t)N: Σt = (1 −t) ̄Σt,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"s = (1 −t) ̄s, we obtain","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"̄Σt( ̄s) = 1 −α −tαd","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"The parameter ̄α(t) := (α −tαd)/(1 −t) now plays the same role as α","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"example that αd < α < αc. Fixing a fraction tc := (αc −α)/(αc −αd)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"ts := (1 −α)/(1 −αd), the total complexity goes to zero, and near-frozen","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"variables become truly frozen, i.e. μi(1) = 0 or 1, as all sub-dominant","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"in N, that is if t < ts. At t = ts, decimation concentrates on a single","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"i, νi({0, 1}) = 1 or νi({0}) = 1 or νi({1}) = 1. For both estimators, any","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"of decimation steps, it outputs νi({0, 1}) = 1 for all i, and the problem can","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"q(a) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"prescriptions (freezings) of B on these aN variables. If t = Θ(N d), the prob-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"a) d1 = x1(α)N: the maximum distance between two solutions inside one","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"b) d2 = x2(α)N: the minimum distance between two solutions from two","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"c) d3 = x3(α)N: the maximum distance between any two solutions (pre-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"w.h.p. by the largest internal entropy sM, i.e. the largest root of Σ(s) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"πA(i) ̸= πB(i). This happens independently with probability p2/2 for each","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"the picture, and its value is ≈0.070. In this figure p = 0.95.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"E[N(x)] = 22(1−α)N","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"if s2(x) := 2(1 −α) −D(x ∥p2/2) > 0, and N(x) = 0 w.h.p. if s2(x) < 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"clusters: x3(α) = 1 −x2(α).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"αs(x) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"where x0 is solution to D(x ∥p2/2) = 2D(x ∥1 −p).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"x2, x3. We put αsep := 1 + (1/2) log(1 −p2/2) > αd, the threshold below","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"We can also define the αgap := αs(x0) > αsep, below which distances","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"For each energy level E0, we define N(E0) = ⌊2NΣ(E0/N)⌋valleys of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"form7 Σ(e0) = a + be0 −ce0 ln(e0), where a > 0 corresponds to a SAT","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"the model by making it energy-dependent: p = p(e0).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"The number N(E0, S0) of valleys of energy E0 = e0N and entropy","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"S0 = s0N is w.h.p.:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"N(E0, S0) ≍2NΣ(e0,s0) if Σ(e0, s0) := Σ(e0) −D(s0 ∥1 −p) ≥0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"E(σ) := min","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq85","equation_number":null,"raw_text":"6 With the notable exception of k-XORSAT, where αd = αsep. Incidentally in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq86","equation_number":null,"raw_text":"k-XORSAT we also have αc = αs.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq87","equation_number":null,"raw_text":"the typical case: for each e0 = E0/N, we compute the distance to the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq88","equation_number":null,"raw_text":"of valleys of state-bottom energy e0N at distance d = ωN is governed by","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq89","equation_number":null,"raw_text":"δ[Σ(e0), p/2]N, where δ(x, y) is solution to: x = D(δ ∥y). Then, the typi-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq90","equation_number":null,"raw_text":"e∗= E/N = min","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq91","equation_number":null,"raw_text":"which shall give us the free energy. Given a valley V of energy E0 = e0N","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq92","equation_number":null,"raw_text":"and entropy S0 = s0N, the number of configurations of energy E belonging","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq93","equation_number":null,"raw_text":"NV (E) = 2S0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq94","equation_number":null,"raw_text":"sV (e|e0, s0) := s0 + (1 −s0)H","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq95","equation_number":null,"raw_text":"where H(x) = −x log2 x−(1−x) log2 (1 −x) is the entropy function. Sum-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq96","equation_number":null,"raw_text":"E = eN is:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq97","equation_number":null,"raw_text":"N(E) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq98","equation_number":null,"raw_text":"s(e) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq99","equation_number":null,"raw_text":"̃s0 = (1 −p)(1 −e + e0)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq100","equation_number":null,"raw_text":"s(e) = max","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq101","equation_number":null,"raw_text":"This will be valid from e = e∗(for which we find s(e∗) = 1 as expected)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq102","equation_number":null,"raw_text":"the border on the definition domain, where Σ(e0, s0) = 0. If we denote by","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq103","equation_number":null,"raw_text":"sM(e0) the biggest valley of energy e0 (i.e. the largest of root of Σ(e0, s0) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq104","equation_number":null,"raw_text":"s(e) = max","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq105","equation_number":null,"raw_text":"temperature is Tc = (∂s/∂e|e=ec)−1. The number of dominating states in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq106","equation_number":null,"raw_text":"ΣT (f) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq107","equation_number":null,"raw_text":"e0,s0:fV (T |e0,s0)=f Σ(e0, s0) ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq108","equation_number":null,"raw_text":"bottom energies and a high state-bottom entropies, which are related by s0 =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq109","equation_number":null,"raw_text":"Σ(e0) = −0.05 −0.5e0 ln e0 and p = 0.2.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq110","equation_number":null,"raw_text":"0) = s∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq111","equation_number":null,"raw_text":"0 = (1−p)(1−e∗+e∗","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq112","equation_number":null,"raw_text":"of a plateau, as it embeds almost all configurations: s(e∗) = 1. Therefore,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq113","equation_number":null,"raw_text":"low configurations of lower energy. This happens at Td := (∂s/∂e|e=e∗)−1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq114","equation_number":null,"raw_text":"described by the microcanonical entropy of a single typical state: sdyn(e) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq115","equation_number":null,"raw_text":"= ∂s","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq116","equation_number":null,"raw_text":"e=e∗= ∂sdyn","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq117","equation_number":null,"raw_text":"e=e∗,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq118","equation_number":null,"raw_text":"Σ(e0) = −0.05 −0.5e0 ln e0 and p = 0.6.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq119","equation_number":null,"raw_text":"arbitrary function of all valleys V and σ; E(σ) = F({V }, σ). By tun-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":50739,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}