structured/0709.0367.structured.json

{"paper_meta":{"paper_id":"arxiv:0709.0367","title":"0709.0367","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:0709.0367v2 [cs.CC] 18 Oct 2007\nProceedings of the International Workshop on Statistical-Mechanical Informatics\nSeptember 16–19, 2007, Kyoto, Japan\nRelationship between clustering and algorithmic\nphase transitions in the random k-XORSAT model\nand its NP-complete extensions\nFabrizio Altarelli1,2, R ́emi Monasson2 and Francesco Zamponi2,3\n1 Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, P.le A. Moro 2, 00185 Roma,\nItaly\n2 CNRS-Laboratoire de Physique Th ́eorique, Ecole Normale Sup ́erieure, 24 rue Lhomond,\n75005 Paris, France\n3 Service de Physique Th ́eorique, Orme des Merisiers, CEA Saclay, 91191 Gif-sur-Yvette\nCedex, France\nE-mail: fabrizio.altarelli@roma1.infn.it\nAbstract.\nWe study the performances of stochastic heuristic search algorithms on Uniquely\nExtendible Constraint Satisfaction Problems with random inputs.\nWe show that, for any\nheuristic preserving the Poissonian nature of the underlying instance, the (heuristic-dependent)\nlargest ratio αa of constraints per variables for which a search algorithm is likely to find solutions\nis smaller than the critical ratio αd above which solutions are clustered and highly correlated.\nIn addition we show that the clustering ratio can be reached when the number k of variables\nper constraints goes to infinity by the so-called Generalized Unit Clause heuristic.\n1. Introduction\nThe application of statistical mechanics ideas and tools to random optimization problems,\ninitiated in the mid-eighties [1], has benefited from a renewed interest from the discovery of\nphase transitions in Constraint Satisfaction Problems (CSP) fifteen years ago. Briefly speaking,\none wants to decide whether a set of randomly drawn constraints over a set of variables admits (at\nleast) one solution. When the number of variables goes to infinity at fixed ratio α of constraints\nper variable the answer abruptly changes from (almost surely) Yes to No when the ratio crosses\nsome critical value αs. Statistical physics studies have pointed out the existence of another\nphase transition in the Yes region [2, 3]. The set of solutions goes from being connected to a\ncollection of disconnected clusters at some ratio αd < αs, a translation in optimization terms of\nthe replica symmetry breaking transition identified by Parisi in mean-field spin glass theory.\nIt is expected that this clustering transition may have dynamical consequences. As replica\nsymmetry breaking signals a loss of ergodicity, sampling algorithms (e.g. Monte Carlo procedure)\nrun into problems at that transition. A quantitative study of the slowing down of MC scheme was\ndone in [4] for the case of the k-XORSAT model where constraints are simply linear equations\n(modulo 2) over k Boolean variables (for an introduction, see [5] and references therein). Yet,\nfinding a solution should in principle be easier than sampling, and the exact nature of the\nrelationship between the performances of resolution algorithms and the static phase transitions\ncharacterizing the solution space is far from being obvious [6]. The present paper is a modest\n\nstep in elucidating this question for the k-XORSAT problem, and some related NP-complete\nproblems sharing the same random structure.\nHereafter we consider simple stochastic search heuristic algorithms working in polynomial\n(linear) time for solving k-XORSAT instances [8, 5].\nBy successively assigning variables\naccording to some heuristic rules those algorithms either produce a solution, or end up with\na contradiction. The probability that a solution is found is a decreasing function of the ratio α,\nand vanishes above some heuristic-dependent ratio αa in the infinite size limit. We show that\nαa < αd for any assignment heuristic in the class of rules preserving the Poissonian structure of\nthe instance. In addition, we determine the most efficient heuristic, that is, the one maximizing\nαa in this class and show that for large k, the two critical ratios match, αa(k) ≃αd(k) ≃log k/k.\nThe plan of the paper is as follows. In section 2 we define the random k-XORSAT decision\nproblem and its extension, as well as the search algorithms studied. Section 3 presents a method\nto characterize the phase diagrams of those random decision problems, depending on the content\n(numbers of constraints over j variables, with j ranging from 1 to k) of their instances. We\nshow that all important information is encoded in a unique ‘thermodynamical’ potential for the\nfraction of frozen variables (backbone). The analysis of the dynamical evolution of the instance\ncontent is exposed in section 4. These dynamical results are combined with the static phase\ndiagram in section 5 to show that the success-to-failure critical ratio of search heuristic, αa, is\nsmaller than the ratio corresponding to the onset of clustering and large backbones, αd. We\nthen show that the so-called Generalized Unit Clause heuristic rule is optimal (in the class of\nPoissonian heuristics) and its critical ratio αa is asymptotically equal to αd in the large k limit.\nOur results are discussed in section 6.\n2. Definitions\n2.1. Decision problems\nThe decision problems we consider in this paper are (k, d)-Uniquely Extendible (UE) Constraint\nSatisfaction Problems (CSP) defined as follows [7]. One considers N variables xi ∈{0, 1, · · · , d−\n1}. A UE constraint, or clause, is a constraint on k variables such that, if one fixes a subset of\nk −1 variables, the value of the k-th variable is uniquely determined. A (k, d)-UE-CSP formula\nis a collection of M = αN clauses, each involving k variables (out of the N available ones). A\nsolution is an assignment of the N variables such that all the clauses are satisfied. k-XORSAT\ncorresponds to d = 2 and is solvable in polynomial time with standard linear algebra techniques.\nFor d = 3 the problem is still in P, while for d ≥4 it has been shown that (3, d)-UE-CSP is\nNP-complete [7].\nA random formula is obtained by choosing, for each clause, the k variables, and the actual\nUE constraint, uniformly at random. It is known that, in the infinite size limit N →∞and at\nfixed clause-to-variable ratio α, [7, 11, 12, 13]:\n• there is a critical ratio αs(k) such that a random (k, d)-UE-CSP is almost surely satisfiable\n(respectively, unsatisfiable) if α < αs(k) (respectively, α > αs(k)).\n• in the satisfiable phase there is another phase transition at some ratio αd(k) such that:\n- for α < αd(k) the space of solutions is ‘connected’: with high probability there is\na path in the set of solutions joining any two solutions such that a step along the path\nrequires to change O(1) variables.\n- for α > αd(k) the space of solution is disconnected into an exponentially large number\nof clusters, each one enjoying the above connectedness property, and far away from each\nother (going from one solution in one cluster to another solution in another cluster requires\nto change O(N) variables). In addition, in each cluster, a finite fraction of variables are\nfrozen i.e. take the same value in all solutions (backbone).\n\n2.2. Search algorithms\nWe will consider simple algorithms acting on the formula in an attempt to find solutions. Those\nalgorithms were introduced and analyzed by Chao and Franco [8] (see [9] for a review). Briefly\nspeaking, starting from a randomly drawn formula, the algorithm assigns one variable at each\ntime step according to the following principles:\n• If there is (at least) one clause of length one (called unit-clause) then satisfy it by adequately\nassigning its variable. This rule is called unit propagation.\n• If all clauses have length two or more, then choose a variable according to some heuristic\nrules. Two simple rules are:\n- Unit Clause (UC): pick up uniformly at random any variable and set it to a random\nuniform value in {0, · · · , d −1};\n- Generalized Unit Clause (GUC): pick up uniformly at random one of the shortest\nclauses, then a variable it this clause, and finally its value.\nIn this analysis, we will discuss a general heuristics in which the variable to be set is chosen among\nthose that appear in the clauses of length j with some probability pj(C1, · · · , Ck), depending in\ngeneral on the number of clauses of length j present in the formula, that we shall call Cj. Unit\npropagation implies that if C1 ̸= 0, then pj = δj,1. We consider also the possibility that the\nvariable is chosen irrespective of the clause length, then Pk\nj=1 pj ≤1.\nBoth UC and GUC are special cases of this general class: in UC variables are chosen at\nrandom, irrespectively of the clauses they appear in (if any), so that pj = 0 unless there are\nunit clauses; GUC corresponds to pj = δj,j∗where j∗is the length of the shortest clause in the\nsystem. Notice that since the variables are selected independently of their number of occurrences,\nthe latter remains Poissonian under the action of the algorithm (even though the value of the\nparameter in the distribution of occurrences may vary). More involved heuristics do exist but\nwill not be analyzed here.\nUnder the action of the algorithm clauses get reduced (decrease in length) until they disappear\nonce satisfied. The algorithm stops either when all clauses have been satisfied or when two\nincompatible unit-clauses have been generated e.g. x = 0 and x = 1. In the latter case the\nalgorithm outputs ‘I do not know whether there is a solution’, while in the former case the\noutput reads ‘Satisfiable’ and returns a solution to the formula. The probability of success,\nthat is, the probability (over the choices of the algorithms and the formula) of getting the\n‘Satisfiable’ output vanishes above some heuristic-dependent ratio αa(< αs) in the infinite N\nlimit. This success-to-failure transition coincides with the polynomial-to-exponential transition\nof backtracking algorithms [5, 10].\n3. ‘Thermodynamical’ Characterization of the Space of Solutions\nUnder the action of the algorithm the length of the clauses changes; therefore the initial (k, d)-\nUE-CSP formula where all clauses have length k evolves into a formula with some distribution\nof clauses of different lengths. We wish then to characterize the space of solutions of a generic\nd-UE-CSP formula made by N variables and by {C0\nj }j=2,···,k clauses of length j, assuming that\nthere are no unit clauses. This characterization will be useful to analyze the performance of\nsearch algorithm in the following.\n3.1. Leaf removal procedure and its analysis\nOur starting observation is that, due to the UE property, when a variable has a unique\noccurrence in the formula, then the clause it appears in can always be satisfied. Hence the\nsubformula obtained by removing this clause is equivalent (in terms of satisfiability) to the\noriginal system [11]. The interest of this remark is that it can be iterated, and more and more\n\nclauses eliminated. Monitoring the evolution of the formula under this procedure, called leaf\nremoval, provides us with useful information on the nature of the solution space [12, 13, 14].\nOne clause is removed at each time step. After T steps we denote by Cj(T) the number of\nclauses of length j. Those numbers obey the evolution equations (in expectation),\nCj(T + 1) −Cj(T) = −\nj Cj(T)\nPk\nj′=2 j′Cj′(T)\n(1)\nwhere the denominator is the total number of occurrences of all variables appearing in the\nformula. The r.h.s. of (1) is simply (minus) the probability that the unique-occurrence variable\nis drawn from a clause of length j.\nIn addition let us define the number Nl(T) of variables appearing in lequations exactly. The\nevolution equations for those numbers are (in expectation)\nNl(T + 1) −Nl(T) =\nk\nX\nj=2\nj(j −1) Cj(T)\nPk\nj′=2 j′Cj′(T)\n×\n (l+ 1) Nl+1(T) −lNl(T)\nP∞\nl′=0 l′Nl′(T)\n \n−δl,1 + δl,0 .\n(2)\nThe above is easy to interpret. The second term in the square bracket on the r.h.s. is the\naverage number of removed variables (other than the single-occurrence variable), that is, the\naverage length of the removed clause minus one. The first term expresses that, if one of those\nvariables appeared l+ 1 times before its removal, the number of its occurrences has decreased\ndown to lafter the removal. Finally, the two δ correspond to the elimination from the system\nof the single-occurrence variable.\nIn the large N limit we may turn those finite difference equations over extensive quantities\nCj, Nlinto differential equations for their intensive counterparts cj = Cj/N, nl= Nl/N as\nfunctions of the reduced number of steps, τ = T/N. The outcome is\ndcj\ndτ\n=\n−jcj\nN ,\n(j = 2, . . . , k) ,\n(3)\ndnl\ndτ\n=\nk\nX\nj=2\nj(j −1)cj\nN\n (l+ 1)nl+1 −lnl\nN\n \n−δl,1 + δl,0 ,\n(4)\nwhere N(τ) = Pk\nj=2 jcj(τ) = P\nl≥1 lnl(τ). The initial conditions are\ncj(0) = C0\nj\nN ;\nnl(0) = e−λ0 (λ0)l\nl!\n,\n(5)\nwhere λ0 is determined by P\nllnl(0) = λ0 = P\nj jcj(0).\nIt is easy to check that equations (3) are solved by cj(τ) = cj(0) b(τ)j provided N\nb\ndb\ndτ = −1.\nIt is convenient to introduce the generating function\nG(b) =\nk\nX\nj=2\ncj(0) bj .\n(6)\nDerivative(s) of G with respect to its argument will be denoted by prime(s). We have that\nN(τ) = b(τ)G′(b(τ)).\nIn addition, we define γ(τ) = P\nj cj(τ) = G(b(τ)).\nWe deduce the\nequation for b(τ):\ndγ\ndτ = N\nb\ndb\ndτ = −1\n⇒\nτ = γ(0) −γ(τ) =\nk\nX\nj=2\ncj(0)(1 −b(τ)j) .\n(7)\n\nThe interpretation of the equation above is just that at each step of the leaf removal one equation\nis eliminated.\nThe solution to (4) remains Poissonian at all times for all l≥2.\nSubstituting nl(τ) =\ne−λ(τ) λ(τ)l\nl!\nwe obtain an equation for λ(τ):\ndλ\ndτ = −\nP\nj≥2 j(j −1)cj(τ)\n(P\nj≥2 jcj(τ))2\nλ(τ) = −\n G′′(b)\nG′(b)2\n \nb=b(τ)\nλ(τ) ,\n(8)\nwith the initial condition imposed by λ(0) = λ0 = P\nj jcj(0) = G′(1).\nFrom (7) we get\ndτ\ndb = −G′(b) so that\ndλ\ndb = dλ\ndτ\ndτ\ndb = G′′(b)\nG′(b) λ ,\n(9)\nwhich is solved by\nλ(b) = G′(b) ,\n(10)\nwhere the normalization is fixed by the initial condition for λ. (7) and (10) determine b(τ) and\nλ(τ), which describe the evolution of the formula under the action of the leaf removal algorithm.\n3.2. Static Phase Transitions\nThe structure of the subformula remaining at the end of the leaf-removal (if any) is indicative\nof the nature of the phase corresponding to typical formulas, uniformly drawn at fixed {C0\nj }.\nThree phases are possible: the unclustered phase where formulas are satisfiable and the solutions\nform a unique cluster; the clustered phase where solutions are divided into many clusters; and\nthe unsat phase where the typical formula is not satisfiable\n(i) Clustering transition:\nThe leaf removal algorithm starts from b = 1, then b decreases\naccording to (7) and the algorithm stops at the largest value of b such that n1 = 0, i.e.\nthere are no more variables with unique occurrence. We have\nn1\n=\nk\nX\nj=2\njcj −\nX\nl>1\nlnl= bG′(b) −\nX\nl>1\nle−λ(b) λ(b)l\nl!\n=\nbλ(b) −e−λ(b)λ(b)\nh\neλ(b) −1\ni\n= λ(b)\nh\nb −1 + e−λ(b)i\n,\ntherefore\nn1 = 0\n⇔\n1 −b = e−λ(b) = e−G′(b) .\n(11)\nThis equation always has the solution b = 0, that gives cj = 0 for all j when the algorithm\nstops. This corresponds to a backbone-free formula whose solution space is connected. On\nthe other hand, if this equation admits non-trivial solutions b > 0, the algorithm stops when\nb is equal to the largest of them, i.e. it is unable to eliminate all clauses in the formula.\nThen the space is clustered and the largest solution represents the fraction of variables in\nthe backbone of each cluster [12, 13].\nIn the pure (k, d)-UE-CSP case, i.e. when c0\nj = αδj,k, the critical ratio at which clustering\nappears decreases with k, from αd(3) ≃0.818 to αd(k) ≃log k/k at large k.\n(ii) Sat/unsat transition: The formula is satisfiable when the subformula left by the removal\nalgorithm has a solution. This happens with high probability if and only if the number of\nequations, given by G(b), is smaller than the number of variables, P\nl≥2 nl[12, 13]. Using\nthe condition n1 = 0, the satisfiability condition is\nG(b) ≤b + (1 −b) log(1 −b) .\n(12)\nFor (k, d)-UE-CSP, the critical ratio at which formulas go from typically satisfiable to\ntypically unsatisfiable increases with k, from αs(3) ≃0.918 to αd(k) →1 at large k.\n\n3.3. The potential for the backbone\nThe outcome of the previous section can be summarized as follows. We considered a formula\nspecified by a set {c0\nj}j=2,···,k, or equivalently by the generating function (6). In the following\nwe will drop the superscript 0 to simplify the notation. We define the potential\nV (b) = −G(b) + b + (1 −b) log(1 −b) .\n(13)\nThe condition n1 = 0 (11), is equivalent to V ′(b) = 0. Thus, if V (b) has a single minimum in\nb = 0, the solution space is not clustered, while if there is another minimum at b ̸= 0, there\nare clusters. Moreover, the condition for satisfiability (12), is that at the secondary minimum\nV (b) ≥0. Examples are given in figure 2.\nThe sat/unsat surface Σs, that separates the sat and the unsat phase, is defined by the\ncondition:\nΣs ≡{cj : V (b) = 0 and V ′(b) = 0 admit a solution b > 0} .\n(14)\nThe clustering surface Σd, that separates the clustered and unclustered regions, is defined\nsimilarly by\nΣd ≡{cj : V ′(b) = 0 and V ′′(b) = 0 admit a solution b > 0} .\n(15)\nThe equations above have to be interpreted as coupled equations for (b, cj); therefore Σs, Σd\nhave dimension k −2 and are surfaces in the space {cj}j=2,···,k of dimension k −1. Note that\nin (14) and (15), one must always choose the largest solution for b, to which we will refer as bs\nand bd, respectively.\nIn addition to the previous sets, in the following a special role will be played by the condition\n2c2 = 1, or equivalently V ′′(0) = 0, that defines the contradiction surface Σq:\nΣq ≡{cj : V ′′(0) = 0} .\n(16)\nThe surface Σq is simply a hyperplane of dimension k −2.\n3.4. The phase diagram\nWe draw a phase diagram in the space of the cj by representing surfaces Σs, Σd, Σq. We focus\non the region cj ∈[0, 1] for j = 3, . . . , k and c2 ∈[0, 1/2]. Indeed, if one of the cj > 1, the\nsystem is surely in the unsat phase [7] while if c2 > 1/2 the algorithm discussed above find a\ncontradiction with very high probability.\nExamples of the phase diagram are in figure 1 for k = 3 and k = 4. There are some special\n“lines” (i.e. intersections of surfaces) on which we will concentrate.\n(i) Recall that Σq is defined by V ′′(0) = 0 and note that V ′(0) = 0 for all b, cj. Thus, on\nΣq, the point b = 0 is a solution of both equations (14) and (15). The surfaces Σs, Σd are\ndefined by the existence of solutions with b > 0, but they might intersect Σq if for some\nvalues of {cj} the solution with b > 0 merges with the solution b = 0. This happen when\nV ′′′(0) = 0, as this is the limiting case in which a saddle at b = bd > 0 and a secondary\nminimum at b = bs > 0 can merge for bd, bs →0. The condition V ′′′(0) = 0 is equivalent to\nc3 = 1/6, and this defines the k −3-dimensional surface\nΣcrit ≡{cj : c2 = 1/2, c3 = 1/6} ,\n(17)\nto which we will refer as critical surface. It is easy to see that the three surfaces Σs, Σd, Σq\nare tangent to each other on the region of the critical surface where they intersect. To\nshow that one must consider a displacement c3 = 1/6 + ε and show that (15), (14) admit a\nsolution with bs, bd ∼ε if c2 −1/2 ∼ε2. We say that in this case the phase transitions are\nof second order because the order parameter b vanishes continuously at the transition.\n\nc\nc\nc2\n3\n4\n1/2\n1/6\n0\n0.2\n0.4\n0.6\n0.8\n1\nc3\n0.2\n0.4\n0.6\n0.8\n1\nc4\n0\n0.1\n0.2\n0.3\n0.4\n0.5\nc2\n0.2\n0.4\n0.6\n0.8\n1\nc3\nFigure 1. (Left) Schematic phase diagram of k=4-UE-CSP. The full (black) curve is the surface\nΣd, the dot-dashed (red) surface is Σs. The two surfaces meet along a portion of the line Σcrit,\ndefined by c2 = 1/2 and c3 = 1/6 and represented as a dashed (blue) line. (Right, top and\nbottom) The sections of Σd (full, black) and of Σs (dot-dashed, red), at fixed c2 (= 0, 0.1, 0.2,\n0.3, 0.4, 0.5 from top to bottom) as a function of c3 on the top panel, and at fixed c4 (= 0, 0.1,\n0.2, 0.3, 0.4, 0.5, 0.6, 0.7 from top to bottom) as a function of c2 in the bottom one. The lines\ncorresponding to c4 = 0 also represent the phase diagram of 3-UE-CSP.\n(ii) There is no a priori reason for which the three surfaces must cross at Σcrit. In fact, the\nsolutions at b > 0 might also disappear discontinuously, like in figure 2, and the surfaces Σs\nand Σd can intersect the surface Σq in regions different from Σcrit. This does not happen\nfor k = 3 but happens for k = 4 for large c4, see figure 1. In this case the transition is called\nfirst order because the order parameter jumps at the transition.\nThe generic phase diagram for all k has the shape of the one for k = 4 which we report in\nfigure 1, left panel.\n4. Search Trajectories in the Space of Formulas\nThe heuristics we defined in section 2 enjoy the property that, after any number of steps of\nthe algorithm, the reduced formula is uniformly distributed over the set of remaining N −T\nvariables conditioned to the numbers Cj(T) of clauses of length j (= 2, ..., k) [8, 9].\nThis\nstatistical property, combined with the concentration phenomenon taking place in the large N\nlimit, allows us to study the evolution of the average clauses densities cj(t) = Cj(T)/N on the\ntime scale t = T/N (fraction of assigned variables), which defines a trajectory in the cj’s space.\nNote that these cj(t) are defined with respect to N, therefore the actual clause density for the\nreduced system of N −T variables are ecj(t) = cj(t)/(1 −t). The trajectory of the ecj(t) moves\nin the cj space of the previous section1.\nInitially we have cj(0) = α δjk, i.e. the evolution starts on the ck axis at ck = α. The\n1 The reader should keep in mind this change of notation to avoid confusion in the following arguments\n\nevolution equation for the densities take the form of first order differential equations,\n ̇cj = (j + 1)cj+1 −jcj\n1 −t\n−ρj(t) .\n(18)\nThe interpretation of the equations above is the following. Let us consider an interval [t, t + dt]\nof continuous time that corresponds to ∆T ∼Ndt time steps of the algorithm. The first term\non the r.h.s. arises from the decrease by one of the length of the clauses that contained the\nvariable just assigned by the algorithms during this interval. The second term corresponds to\nan additional loss of clauses which is present when the variable is selected from a clause of\nlength j: as the heuristics explicitly chooses an equation (and a variable therein) of length j\nwith probability pj (see section 2), this equation will be reduced irrespectively of the number of\nother occurrences of the variable. Hence ρj(t) is given, for j ≥1, by\nρj(t) =\nlim\n∆T→∞lim\nN→∞\n1\n∆T\ntN+∆T−1\nX\nT=tN\n(pj −pj+1) ≡⟨pj −pj+1⟩,\n(19)\nwhere both pj, pj+1 depend on their arguments (numbers of clauses) and ⟨•⟩represents the\naverage over ∆T defined in (19). Here pk+1 ≡0. Note that the case j = 1 is special as all\nclauses of length one that are produced are immediately eliminated. On average\nρ1 ≡2c2\n1 −t\n(20)\nclauses of length 2 become of length 1 and are then eliminated by unit propagation. The total\nfraction of eliminated clauses is\n ̇γ(t) ≡−\nk\nX\nj=2\n ̇cj(t) = 2c2(t)\n1 −t +\nk\nX\nj=2\nρj(t) =\nk\nX\nj=1\nρj(t) ≤1 ,\n(21)\nwhere the last inequality follows from (19). As only clauses of length one are eliminated, the\nviolation of (21) can only happen if too many such clauses are generated. This corresponds\nto ρ1 →1−; in this case a contradiction occurs with high probability and the algorithm stops\nwith the ‘Don’t know’ output. When ρ1 →1−, the algorithm makes only unit propagations and\nρj →0+ for all j ≥2. For this reason we called the plane ρ1 = 1, i.e. ec2 = 1/2, contradiction\nsurface.\n4.1. Unit Clause (UC)\nIn the UC heuristic variables are chosen at random when there is no unit clause. Hence ρj = 0\nfor j = 2, · · · , k. The solution to (18) is cj(t) = α\n k\nj\n (1 −t)jtk−j. The algorithm will generate\na contradiction with high probability (w.h.p.) if the average number of unit clauses starts to\nbuild-up, i.e. if 2c2(t)/(1 −t) ≥1. This gives an equation for the value of α at which the\nprobability that the algorithm finds a solution vanishes: for k = 3, α(UC)\na\n= 2/3.\n4.2. Generalized Unit Clause (GUC)\nIn the GUC heuristic the algorithm always fixes a variable appearing in the shortest clauses. In\nthe continuum limit cj = 0 for j smaller than a given value; therefore we define\nj∗(t) = min{j : cj(t) > 0} ,\n(22)\n\nthe minimal length of clauses with positive densities. We also define\nt∗(j) = min[t : cj−1(t) > 0]\n(23)\nthe time at which j∗jumps down from j to j −1. Essentially, the algorithm picks one clause of\nlength j∗and assigns successively all the variables in this clause until the clause disappears. But\nin doing so, other clauses of length j < j∗are generated and have to be eliminated to recover\nthe situation in which Cj = 0 for all j < j∗; for this reason ρj∗is not given exactly by 1/j∗.\nWhen the number of generated clauses is so high that the algorithm is unable to remove them,\ncj∗−1 becomes different from 0 and j∗jumps down by 1. The resulting motion equations for the\nclause densities are, for j ≥j∗(t):\n ̇cj(t) = (j + 1)cj+1(t) −jcj(t)\n1 −t\n−δj,j∗(t)\n 1\nj −(j −1)cj(t)\n1 −t\n \n.\n(24)\nThe transition times t∗are given by\ncj(t∗(j))\n1 −t\n=\n1\nj(j −1) ,\n(25)\nwhere the algorithm is no more able to remove the clauses of length j∗because too many clauses\nof length j∗−1 are being generated by propagations.\nComparing with (18) above, we observe that in the interval t ∈[t∗(j+1), t∗(j)], where j∗= j,\nonly two ρj are different from 0:\nρj∗= 1\nj∗−(j∗−1)cj∗(t)\n1 −t\n,\nρj∗−1 = j∗cj∗(t)\n1 −t\n,\n(26)\nthe first representing clauses of length j∗which are directly eliminated, the second representing\nthe clauses of length j∗−1 that are produced and subsequently eliminated in the process. In\nthis interval of time, the ratio cj∗(t)/(1 −t) increases from 0 to 1/j∗/(j∗−1) from condition\n(25). Then\n1\nj∗(t) ≤ ̇γ(t) = (ρj∗+ ρj∗−1) ≤\n1\nj∗(t) −1 ,\n(27)\nwhich is consistent with (but stronger than) (21) above.\n5. Analysis of the “dynamic” phase diagram\nConsider now a given heuristic, and a generic (k, d)-UE-CSP formula specified by its clause-to-\nvariable ratio α. The formula, in the cj space, starts on the axis ck at ck = α. The evolution\nof the formula under the action of the algorithm is represented by a trajectory {cj(t, α)}j=2,···,k\nor equivalently by G(b; t, α) = Pk\nj=2 bjcj(t, α), that depends on α through the initial condition\nG(b; 0, α) = αbk. We define a potential V (b; t, α) by replacing in (13) G(b) →G(b; t, α)/(1 −t);\nthe normalization (1 −t) is due to the fact that the cj = Cj/N are divided by N instead of\nN −T.\nWe follow the evolution of the formula by looking at the times at which the trajectory\nstarting at ck = α at time 0 crosses the surfaces Σs, Σd, Σq defined in section 3.3, which we call\nts(α), td(α), tq(α) respectively. As an example, in figure 2 we report the potential at different\ntimes during the evolution of a formula according to the UC heuristic for α > α(UC)\na\n.\nWe draw a “dynamic phase diagram” by representing in the (t, α) plane the lines separating\nthe unclustered, clustered, unsat and contradiction phases, which we call αd(t), αs(t), αq(t) and\n\n0\n0.2\n0.4\n0.6\n0.8\n1\nb\n0\n0.05\n0.1\n0.15\n0.2\nV(b)\nFigure 2.\nAn example of the potential V (b; t, α) plotted (from top to bottom) at times\nt = {0, td = 0.02957, 0.07327, ts = 0.11697, 0.20642} during the evolution of a (3, d)-UE-CSP\nformula with α = 0.8 under the UC heuristic. In the unclustered region it is a convex function\nof b with a global minimum in b = 0. On the clustering line td it first develops a secondary\nminimum. On the sat/unsat line the value of V at the secondary minimum becomes equal to 0.\nare just the inverse of the times defined above.\nExamples in the case of the UC and GUC\nheuristics are given in figure 3.\nFrom the general properties of the function V (b; t, α) we can deduce a number of properties\nof the lines αd(t), αs(t), αq(t). We will show that the three lines intersect at a “critical point”\n(ta, αa), located at αa ≤αd, under the more general conditions. This implies that the algorithm\nstops working at the value αa ≤αd, which is our central result: Poissonian search algorithm\ncannot find a solution in polynomial time in the clustered region.\n5.1. Equations for the transition lines\nThe generating function G(b; t, α) satisfy an evolution equation which is easily derived from (18):\n ̇G(b; t, α)\n=\n1 −b\n1 −tG′(b; t, α) −F(b; t, α) ,\n(28)\nF(b; t, α)\n≡\n2c2(t)\n1 −t b +\nk\nX\nj=2\nρj(t)bj =\nk\nX\nj=1\nρj(t)bj .\n(29)\nPerforming the total derivative with respect to t of the first condition (V ′ = 0) in (15) for\n(αd, bd) and using the second condition, V ′′ = 0, we have\n∂V ′\n∂α\ndαd\ndt + ̇V ′ = 0\n⇒\ndαd\ndt = −\n ̇V ′(bd; t, αd)\n∂V ′\n∂α (bd; t, αd) .\n(30)\nUsing the definition (13) we have\n ̇V ′(b; t, α)\n=\n−\n1\n1 −t\n \n ̇G′(b; t, α) + G′(b; t, α)\n1 −t\n \n,\n(31)\n\n0.6\n0.7\n0.8\n0.9\n1\nα\n0\n0.2\n0.4\n0.6\nt\nαd\nαs\nαq\nUC\nGUC\nα\nt\nFigure 3. (Left) Phase boundary lines in the (t, α) plane for the UC and GUC heuristics for\nk = 3. The three lines meet at the critical point (ta, αa) at which the algorithm is no more\nable to find a solution (black dot). (Right) The generic shape of the clustering and of the\nsat/unsat lines. The possibility of a maximum cannot be excluded, but in any case t must be a\nsingle-valued function of α, meaning that if the algorithm enters the cluster (or unsat) phase it\ncannot escape at later times.\n∂V ′\n∂α\n=\n−\n1\n1 −t\n∂G′\n∂α = −\n1\n1 −t\nX\nj≥2\njbj−1 ∂cj(t, α)\n∂α\n.\n(32)\nThen\ndαd\ndt = −\n ̇G′(b; t, α) + G′(b;t,α)\n1−t\n∂αG′(b; t, α)\n \nα=αd(t),b=bd(t)\n.\n(33)\nUsing (28) and differentiating with respect to b we have\n ̇G′(b; t, α) + G′(b; t, α)\n1 −t\n= 1 −b\n1 −tG′′(b; t, α) −F ′(b; t, α) .\n(34)\nNow using V ′′(b; t, α) = −G′′(b;t,α)\n1−t\n+\n1\n1−b and V ′′(bd, t) = 0 we have 1−b\n1−tG′′(b; t, α) = 1 for b = bd\nand finally we get\ndαd\ndt = −1 −F ′(b; t, α)\n∂αG′(b; t, α)\n \nα=αd(t),b=bd(t)\n.\n(35)\nA very similar reasoning leads to the following equation for the sat/unsat line:\ndαs\ndt = −b −F(b; t, α)\n∂αG(b; t, α)\n \nα=αs(t),b=bs(t)\n.\n(36)\nThe equation for the contradiction line is easily derived from its definition ec2(t, α) = c2(t,α)\n1−t\n= 1\n2,\nwhich immediately gives\ndαq\ndt = −1 + 2 ̇c2(t, α)\n2∂αc2(t, α)\n \nα=αq(t)\n.\n(37)\n\n5.2. General properties of the transition lines\nWe wish to show that the transition lines td(α),ts(α) and tq(α) in the (α, t) plane are single-\nvalued functions of α, and that they meet in a point (αa, ta) where they have infinite slope and\nare therefore tangent to each other; the value αa correspond to a trajectory which is tangent to\nthe crytical surface Σcrit.\nOur argument goes as follows:\n(i) We defined αa as the value of α for which the probability of finding a solution for the\nchosen heuristic vanishes. Then the trajectory2 corresponding to any α > αa must cross\nthe contradiction surface, while the trajectory corresponding to any α < αa must not cross\nit, so that the trajectory corresponding to αa must be tangent to the contradiction surface\nΣq. The latter trajectory is tangent to Σq when ec2(t) = 1/2,\nd\ndtec2(t) = 0; the solution to\nthese conditions gives ta and αa.\nMoreover, ec2(t) = 1/2 implies that ρ1 = 1 which then implies ρj = 0 for all j ≥2, as already\ndiscussed. Then we have\nd\ndtec2(t) = d\ndt\n2c2(t)\n1 −t = 2 ̇c2(t)\n1 −t + 2c2(t)\n(1 −t)2 = 0\n⇒\n ̇c2(t) = −c2(t)\n1 −t = −1\n2 ,\n(38)\nwhich, together with the equations of motion (18) and ρ2 = 0 gives\n−c2(t)\n1 −t = dc2(t)\ndt\n= 3c3(t) −2c2(t)\n1 −t\n⇒\nec3(t) = c3(t)\n1 −t = 1\n3\nc2(t)\n1 −t = 1\n6 .\n(39)\nTherefore the point where the trajectory for α = αa is tangent to the contradiction surface\nbelongs to the critical surface Σcrit. From equation (37) it is clear that since ̇c2 = −1/2,\nthe function tq(α) has infinite slope in (ta, αa), as in figure 3.\n(ii) Next we show that the numerators of the fractions appearing in ̇αd(t) and ̇αs(t) are strictly\npositive if t < tq(α), i.e. in before a contradiction is found. Using the definition (19) we\ncan write:\nF(b; t, α)\n=\nk\nX\nj=1\nρj(t)bj = b\n \n ⟨p1⟩+\nk\nX\nj=2\nbj−2(b −1) ⟨pj⟩\n \n ,\n(40)\nF ′(b; t, α)\n=\nk\nX\nj=1\njρj(t)bj−1 = ⟨p1⟩+\nk\nX\nj=2\nbj−2 [1 −j(1 −b)] ⟨pj⟩.\nThe coefficients in front of ⟨pj⟩≥0 in the sums above are always smaller than 1,\nindependently of j, so that\nF(b; t, α)\n≤\nb\n \n ⟨p1⟩+\nk\nX\nj=2\n⟨pj⟩\n \n ≤b ,\n(41)\nF ′(b; t, α)\n≤\n⟨p1⟩+\nk\nX\nj=2\n⟨pj⟩≤1 .\n(42)\nThe functions F(b; t, α) and F ′(b; t, α) are to be computed in b = bs(t, α) or b = bd(t, α)\nin equations (35) and (36). Both bs and bd are strictly smaller than 1 for all (t, α), as one\ncan directly show from their definitions because V ′(b →1) →∞. Then the coefficients in\nthe sums in (40) are strictly smaller than 1, and the only solution to F = b or F ′ = 1 is\n⟨pj⟩= δ1j, which happens only on the contradiction line.\n2 Recall that we are here talking about average trajectories.\n\n(iii) The denominators in equations (35), (36) are surely positive at t = 0, as G(b; 0, α) = αbk\nindependently of the heuristic. If they remain positive at all times, then ̇αd(t), ̇αs(t) ≤0 at\nall times, or equivalently dtd\ndα , dts\ndα ≤0 at all α, so that td, ts always increase on decreasing α.\nThe other possibility is that the denominator in (35) crosses zero and become negative,\nleading to a maximum in td(α), which will then decrease on decreasing α. Possibly the\ndenominators can vanish again, giving rise to a sequence of maxima and minima, see right\npanel of figure 3.\nWhat is important is that the numerator is always strictly positive, and as a consequence\ntd(α) or ts(α) are single-valued functions of α. In fact, for td(α) or ts(α) to be multiple-\nvalued functions of α, at some point their slope must become infinite, which is excluded by\nthe analysis above.\n(iv) The statement above, that td(α) and ts(α) are single valued functions of α, implies that\nif a trajectory enters the clustered or unsat phase, it cannot exit from it. This is enough\nto show that αa ≤αd; in fact, the trajectory for α = αa cannot start inside the clustered\nphase, as it would not be able to escape and reach the origin, which is required to find a\nsolution.\n(v) In general the function ec2(t) increases until it reaches a maximum and then decreases\nto 0.\nFor α = αa the value at the maximum is ec2 = 1/2.\nFor α > αa, the value\nat the maximum is ec2 > 1/2, therefore the contradiction ec2 = 1/2 is reached before\nthe maximum, when ec2 is still increasing.\nThen\nd\ndtec2 > 0 at the contradiction point.\nPerforming a simple computation similar to equations (38), (39), one can show that the\ntrajectories for α > αa meet the contradiction surface at ec3 > 1/6.\nNotice then that,\nas it is evident in figure 1, the trajectories corresponding to α > αa must enter first the\nclustered and then the unsat phases in order to reach the contradiction surface, therefore\nfor α < αa one has td(α) < ts(α) < tq(α). On the contrary the trajectories corresponding\nto α < αa must stay away from the clustering and sat/unsat surfaces, otherwise they could\nnot exit and should meet the contradiction surface: therefore for any α < αa, td(α) and\nts(α) do not exist. For α →α+\na , as the surfaces Σd, Σs, Σq are tangent in Σcrit, one has\ntd(αa) = ts(αa) = tq(αa) = ta and the three curves have infinite slope as all the numerators\nin equations (35), (36), (37) vanish on the contradiction surface. This is indeed what is\nobserved in figure 3 for the UC and GUC heuristics, and this argument confirms that this\nis the generic behavior for all the heuristics in the class considered here.\nThis structure is particularly evident for UC, where\nG(UC)(b; t, α) = α[1 −(1 −b)(1 −t)]k −αtk−1[kb(1 −t) + t] .\n(43)\nFrom (43) it is straightforward to check that ∂αG(b; t, α) > 0, ∂αG′(b; t, α) > 0, if b > 0. Then,\nas F(b; t, α) = 2bc2(t)\n1−t\nfor UC, both ̇αd(t) and ̇αs(t) are proportional to 2c2(t)\n1−t −1. This means\nthat αs,αd are decreasing functions of t below the contradiction line.\nThe conclusion is that for a generic Poissonian heuristic, the three lines cross at a critical\npoint (ta, αa) which depends on the heuristic. Above αa the heuristic will cross all the lines\nand find a contradiction. From the properties of the dynamical line, we have that generically\nαa ≤αd, that is no Poissonian search heuristic can find a solution in polynomial time above\nαd, as stated at the beginning of this section. The natural question is then if there exists an\nheuristic that saturates the bound, i.e. such that αa = αd. From the discussion above it is clear\nthat this is possible only if ̇αd(t) ≡0, i.e. the dynamical line in the (t, α) plane is a straight\nvertical line, which is possible only if the numerator in (35) is identically vanishing.\n\n5.3. Optimality of GUC\nIt is quite easy to see that GUC is the heuristic that locally optimizes the numerator in (35).\nIndeed, from the definition F ′(b; t, α) = Pk\nj=1 jbj−1ρj and the bound F ′(1, t) ≤1, it is clear\nthat F ′(b; t, α) is maximized by maximizing ρj for the smallest possible j, i.e. by picking clauses\nfrom the shortest possible ones, that is GUC. Unfortunately a general proof of the optimality\nof GUC for finite k seems difficult, because one should prove that GUC optimizes globally the\nclustering line, and also control the denominator in (35). In this section we will show that for\nk →∞, GUC is optimal in the sense that ̇αd ≡0 and αd = αa at leading order in k.\nFrom the definition γ(t) = −Pk\nj=2 cj(t) and integrating over time the bound (27), we have\nfor GUC:\nα −\nZ t\n0\ndt′\nj∗(t′) −1 ≤−γ(t) ≤α −\nZ t\n0\ndt′\nj∗(t′) .\n(44)\nor, equivalently,\nα −\nX\nj\nt∗(j) −t∗(j + 1)\nj −1\n≤−γ(t) ≤α −\nX\nj\nt∗(j) −t∗(j + 1)\nj\n.\n(45)\nwhere the sums are limited to the values of j that are reached during the search. In the large k\nlimit, provided the hypothesis\nt∗(j) −t∗(j + 1) = 1\nk + o(1/k)\n(46)\nholds for most j, we obtain\n−γ(t) ≃α −1\nk\nk\nX\nj∗(t)\n1\nj .\n(47)\nThe hypothesis (46) is well supported by numerical data, as shown in figure 4. As the sum of the\ninverse of the first k integers is equivalent to log k (harmonic number) we see that the minimal\nvalue of j∗over t is much larger than 2 if α is much smaller than log k/k. Therefore\nαa ≥log k\nk\n.\n(48)\nThe r.h.s. of the above inequality coincides with the asymptotic scaling of the clustering critical\nratio (section 3.2). Since the results of the previous section require that αa ≤αd, we obtain\nthat α(GUC)\na\n= αd ≃log k/k at the leading order in k →∞. As a comparison, it is easy to see\nthat for UC the threshold for large k is α(UC)\na\n≃e/k, which is therefore much lower than the\nthreshold for GUC.\nThese arguments are supported by numerical simulations that we performed up to k = 216,\nin which the equations of motion (24) are integrated as finite differences equations for all values\nof j (see figure 4). The numerical investigation confirms that kα(GUC)\na\nis very well fitted by\nlog k + 2.15 for k in the range 28 ÷ 216. Moreover, a finite size scaling analysis (with respect to\nk) of the data shown in figure 4 shows that\nk[t∗(j) −t∗(j + 1)] = 1 + kν × f(j/k)\n(49)\nwhere f(x) is a function independent on k which behaves as x−μ for x close to 0. From the\nnumerical data, it appears that ν = μ = 0.5, which confirms that the first correction to the\nleading term log k/k is of order 1/k.\n\n0\n0.2\n0.4\n0.6\n0.8\n1\nj k\n0.7\n0.8\n0.9\n1\n1.1\n1.2\n1.3\nk Dt*\n5\n6\n7\n8\n9\n10\n11\n12\nlogHkL\n7\n8\n9\n10\n11\n12\n13\n14\nk Αa\n0\n0.2\n0.4\n0.6\n0.8\n1\nj k\n-7.5\n-5\n-2.5\n0\n2.5\n5\n7.5\n10\n !!!k Hk Dt - 1L\n-5\n-4\n-3\n-2\n-1\nlogHxL\n0.5\n1\n1.5\n2\n2.5\n3\nlogI !!!k Hk Dt - 1LM\nFigure 4. Finite size scaling results for GUC at large k. Top Left Each curve shows the values\nof k[t∗(j) −t∗(j + 1)] as a function of j/k for k = 28, 29, . . . , 216 (from the farthest to the closest\ncurve to 1), and was obtained by integrating the equations of motion (24) by finite differences.\nFor each k, the value of α used is αGUC\na\n(k), determined as the value of α for which the maximum\nreached by 2c2(t)/(1 −t) is 1. Top Right Data points of αGUC\na\n(k) versus log k/k + 2.15/k (full\nred line). Bottom left The same data as above, plotted as {k × [t∗(j) −t∗(j + 1)]} × k1/2. The\ncurves “collapse”, showing f(x) and confirming the value of ν = 1/2. Bottom right By plotting\nthe same curves on logarithmic scale it is easily seen that for x close to 0 f(x) ≃x−μ with\nμ = 1/2, corresponding to the slope of the full red line.\n6. Conclusions\nOne of the main results of this paper, that is, that linear-time search heuristic are not able to\nsolve instances in the clustered phase of UE-CSP problems should be interpreted with care. In\nXORSAT-like models the clustering transition coincide with the emergence of strong correlations\nbetween variables in solutions, while the two phenomena generally define two distinct critical\nratios for other random decision problems [15, 16]. From an intuitive point of view it is expected\nthat the performances of search heuristics are affected by correlations between variables rather\nthan the clustering of solutions. Indeed, as the search algorithms investigated here do not allow\nfor backtracking or corrections of wrongly assigned variables, very strong correlations between\nO(N) variables (recall that the backbone includes O(N) variables in the clustered phase) are\nlikely to result in e−O(N) probabilities of success for the algorithm.\nExtending the present work to the random Satisfiability (k-SAT) problem would be interesting\nfrom this point of view, because even if the clustering and freezing transition coincide at leading\n\norder for k →∞[3], their finite k values are different in this case. Moreover, in some similar\nproblems (k-COL [17] and 1-in-k-SAT [18]) it has been proven that search algorithms similar to\nthe ones investigated here are efficient beyond the point where the replica-symmetry-breaking\nsolution is stable.\nTherefore these algorithms might beat the clustering threshold in these\nproblems. Note however that in these cases the transition is continuous, so that the structure\nof the clusters is expected to be very different from the one of XORSAT.\nIn addition, while the Generalized Unit Clause heuristic is here shown to be optimal for the\nk-XORSAT problem and to saturate the clustering ratio when k →∞, it is certainly not the\ncase of the k-SAT problem. Determining a provably optimal search heuristic for this problem\nremains an open problem.\n[1] M ́ezard M, Parisi G and Virasoro M A 1987 Spin glass theory and beyond (Singapore: World Scientific)\n[2] Biroli G, Monasson R and Weigt M 2000 Eur. Phys. J. B 14 551\n[3] Krzakala F, Montanari A, Ricci-Tersenghi F, Semerjian G and Zdeborov ́a L 2007 Proc. Natl. Acad. Sci. USA\n104 10318\n[4] Montanari A and Semerjian G 2006 J. Stat. Phys. 124 103\n[5] Monasson R 2007 Introduction to Phase Transitions in Random Optimization Problems, Lecture Notes of\nthe Les Houches Summer School on Complex Systems, Elsevier\n[6] Krzakala F and Kurchan J 2007 Phys. Rev. E 76 021122\n[7] Connamacher H 2004 A Random Constraint Satisfaction Problem That Seems Hard for DPLL, Proceedings\nof the Seventh International Conference on Theory and Applications of Satisfiability Testing\n[8] Chao M T and Franco J 1990 Information Science 51 289\nChao M T and Franco J 1986 SIAM Journal on Computing 15 1106\n[9] Achlioptas D 2001 Theor. Comp. Sci. 265 159\n[10] Achlioptas D, Beame P and Molloy M 2004 J. Comput. Syst. Sci. 68 238\n[11] Dubois O and Mandler J 2002 The 3-XORSAT Threshold, Proceedings of the 43rd Symposium on Foundations\nof Computer Science\n[12] M ́ezard M, Ricci-Tersenghi F and Zecchina R, 2003 J. Stat. Phys. 111 505\n[13] Cocco S, Dubois O, Mandler J and Monasson R 2003 Phys. Rev. Lett. 90 047205\n[14] Weigt M 2002 Eur. Phys. J. B 28 369\n[15] Semerjian G 2007 On the freezing of variables in random constraint satisfaction problems Preprint\narXiv:cond-mat/07052147 (J.Stat.Phys. in press)\n[16] Krzakala F and Zdeborov ́a L 2007 Phys. Rev. E 76 031131\n[17] Achlioptas D and Moore C 2003 J. Comput. Syst. Sci. 67 441\n[18] Raymond J, Sportiello A and Zdeborov ́a L 2007 Phys. Rev. E 76 011101","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0709.0367v2 [cs.CC] 18 Oct 2007\nProceedings of the International Workshop on Statistical-Mechanical Informatics\nSeptember 16–19, 2007, Kyoto, Japan\nRelationship between clustering and algorithmic\nphase transitions in the random k-XORSAT model\nand its NP-complete extensions\nFabrizio Altarelli1,2, R ́emi Monasson2 and Francesco Zamponi2,3\n1 Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, P.le A. Moro 2, 00185 Roma,\nItaly\n2 CNRS-Laboratoire de Physique Th ́eorique, Ecole Normale Sup ́erieure, 24 rue Lhomond,\n75005 Paris, France\n3 Service de Physique Th ́eorique, Orme des Merisiers, CEA Saclay, 91191 Gif-sur-Yvette\nCedex, France\nE-mail: fabrizio.altarelli@roma1.infn.it\nAbstract.\nWe study the performances of stochastic heuristic search algorithms on Uniquely\nExtendible Constraint Satisfaction Problems with random inputs.\nWe show that, for any\nheuristic preserving the Poissonian nature of the underlying instance, the (heuristic-dependent)\nlargest ratio αa of constraints per variables for which a search algorithm is likely to find solutions\nis smaller than the critical ratio αd above which solutions are clustered and highly correlated.\nIn addition we show that the clustering ratio can be reached when the number k of variables\nper constraints goes to infinity by the so-called Generalized Unit Clause heuristic.\n1. Introduction\nThe application of statistical mechanics ideas and tools to random optimization problems,\ninitiated in the mid-eighties [1], has benefited from a renewed interest from the discovery of\nphase transitions in Constraint Satisfaction Problems (CSP) fifteen years ago. Briefly speaking,\none wants to decide whether a set of randomly drawn constraints over a set of variables admits (at\nleast) one solution. When the number of variables goes to infinity at fixed ratio α of constraints\nper variable the answer abruptly changes from (almost surely) Yes to No when the ratio crosses\nsome critical value αs. Statistical physics studies have pointed out the existence of another\nphase transition in the Yes region [2, 3]. The set of solutions goes from being connected to a\ncollection of disconnected clusters at some ratio αd < αs, a translation in optimization terms of\nthe replica symmetry breaking transition identified by Parisi in mean-field spin glass theory.\nIt is expected that this clustering transition may have dynamical consequences. As replica\nsymmetry breaking signals a loss of ergodicity, sampling algorithms (e.g. Monte Carlo procedure)\nrun into problems at that transition. A quantitative study of the slowing down of MC scheme was\ndone in [4] for the case of the k-XORSAT model where constraints are simply linear equations\n(modulo 2) over k Boolean variables (for an introduction, see [5] and references therein). Yet,\nfinding a solution should in principle be easier than sampling, and the exact nature of the\nrelationship between the performances of resolution algorithms and the static phase transitions\ncharacterizing the solution space is far from being obvious [6]. The present paper is a modest"},{"paragraph_id":"p2","order":2,"text":"step in elucidating this question for the k-XORSAT problem, and some related NP-complete\nproblems sharing the same random structure.\nHereafter we consider simple stochastic search heuristic algorithms working in polynomial\n(linear) time for solving k-XORSAT instances [8, 5].\nBy successively assigning variables\naccording to some heuristic rules those algorithms either produce a solution, or end up with\na contradiction. The probability that a solution is found is a decreasing function of the ratio α,\nand vanishes above some heuristic-dependent ratio αa in the infinite size limit. We show that\nαa < αd for any assignment heuristic in the class of rules preserving the Poissonian structure of\nthe instance. In addition, we determine the most efficient heuristic, that is, the one maximizing\nαa in this class and show that for large k, the two critical ratios match, αa(k) ≃αd(k) ≃log k/k.\nThe plan of the paper is as follows. In section 2 we define the random k-XORSAT decision\nproblem and its extension, as well as the search algorithms studied. Section 3 presents a method\nto characterize the phase diagrams of those random decision problems, depending on the content\n(numbers of constraints over j variables, with j ranging from 1 to k) of their instances. We\nshow that all important information is encoded in a unique ‘thermodynamical’ potential for the\nfraction of frozen variables (backbone). The analysis of the dynamical evolution of the instance\ncontent is exposed in section 4. These dynamical results are combined with the static phase\ndiagram in section 5 to show that the success-to-failure critical ratio of search heuristic, αa, is\nsmaller than the ratio corresponding to the onset of clustering and large backbones, αd. We\nthen show that the so-called Generalized Unit Clause heuristic rule is optimal (in the class of\nPoissonian heuristics) and its critical ratio αa is asymptotically equal to αd in the large k limit.\nOur results are discussed in section 6.\n2. Definitions\n2.1. Decision problems\nThe decision problems we consider in this paper are (k, d)-Uniquely Extendible (UE) Constraint\nSatisfaction Problems (CSP) defined as follows [7]. One considers N variables xi ∈{0, 1, · · · , d−\n1}. A UE constraint, or clause, is a constraint on k variables such that, if one fixes a subset of\nk −1 variables, the value of the k-th variable is uniquely determined. A (k, d)-UE-CSP formula\nis a collection of M = αN clauses, each involving k variables (out of the N available ones). A\nsolution is an assignment of the N variables such that all the clauses are satisfied. k-XORSAT\ncorresponds to d = 2 and is solvable in polynomial time with standard linear algebra techniques.\nFor d = 3 the problem is still in P, while for d ≥4 it has been shown that (3, d)-UE-CSP is\nNP-complete [7].\nA random formula is obtained by choosing, for each clause, the k variables, and the actual\nUE constraint, uniformly at random. It is known that, in the infinite size limit N →∞and at\nfixed clause-to-variable ratio α, [7, 11, 12, 13]:\n• there is a critical ratio αs(k) such that a random (k, d)-UE-CSP is almost surely satisfiable\n(respectively, unsatisfiable) if α < αs(k) (respectively, α > αs(k)).\n• in the satisfiable phase there is another phase transition at some ratio αd(k) such that:\n- for α < αd(k) the space of solutions is ‘connected’: with high probability there is\na path in the set of solutions joining any two solutions such that a step along the path\nrequires to change O(1) variables.\n- for α > αd(k) the space of solution is disconnected into an exponentially large number\nof clusters, each one enjoying the above connectedness property, and far away from each\nother (going from one solution in one cluster to another solution in another cluster requires\nto change O(N) variables). In addition, in each cluster, a finite fraction of variables are\nfrozen i.e. take the same value in all solutions (backbone)."},{"paragraph_id":"p3","order":3,"text":"2.2. Search algorithms\nWe will consider simple algorithms acting on the formula in an attempt to find solutions. Those\nalgorithms were introduced and analyzed by Chao and Franco [8] (see [9] for a review). Briefly\nspeaking, starting from a randomly drawn formula, the algorithm assigns one variable at each\ntime step according to the following principles:\n• If there is (at least) one clause of length one (called unit-clause) then satisfy it by adequately\nassigning its variable. This rule is called unit propagation.\n• If all clauses have length two or more, then choose a variable according to some heuristic\nrules. Two simple rules are:\n- Unit Clause (UC): pick up uniformly at random any variable and set it to a random\nuniform value in {0, · · · , d −1};\n- Generalized Unit Clause (GUC): pick up uniformly at random one of the shortest\nclauses, then a variable it this clause, and finally its value.\nIn this analysis, we will discuss a general heuristics in which the variable to be set is chosen among\nthose that appear in the clauses of length j with some probability pj(C1, · · · , Ck), depending in\ngeneral on the number of clauses of length j present in the formula, that we shall call Cj. Unit\npropagation implies that if C1 ̸= 0, then pj = δj,1. We consider also the possibility that the\nvariable is chosen irrespective of the clause length, then Pk\nj=1 pj ≤1.\nBoth UC and GUC are special cases of this general class: in UC variables are chosen at\nrandom, irrespectively of the clauses they appear in (if any), so that pj = 0 unless there are\nunit clauses; GUC corresponds to pj = δj,j∗where j∗is the length of the shortest clause in the\nsystem. Notice that since the variables are selected independently of their number of occurrences,\nthe latter remains Poissonian under the action of the algorithm (even though the value of the\nparameter in the distribution of occurrences may vary). More involved heuristics do exist but\nwill not be analyzed here.\nUnder the action of the algorithm clauses get reduced (decrease in length) until they disappear\nonce satisfied. The algorithm stops either when all clauses have been satisfied or when two\nincompatible unit-clauses have been generated e.g. x = 0 and x = 1. In the latter case the\nalgorithm outputs ‘I do not know whether there is a solution’, while in the former case the\noutput reads ‘Satisfiable’ and returns a solution to the formula. The probability of success,\nthat is, the probability (over the choices of the algorithms and the formula) of getting the\n‘Satisfiable’ output vanishes above some heuristic-dependent ratio αa(< αs) in the infinite N\nlimit. This success-to-failure transition coincides with the polynomial-to-exponential transition\nof backtracking algorithms [5, 10].\n3. ‘Thermodynamical’ Characterization of the Space of Solutions\nUnder the action of the algorithm the length of the clauses changes; therefore the initial (k, d)-\nUE-CSP formula where all clauses have length k evolves into a formula with some distribution\nof clauses of different lengths. We wish then to characterize the space of solutions of a generic\nd-UE-CSP formula made by N variables and by {C0\nj }j=2,···,k clauses of length j, assuming that\nthere are no unit clauses. This characterization will be useful to analyze the performance of\nsearch algorithm in the following.\n3.1. Leaf removal procedure and its analysis\nOur starting observation is that, due to the UE property, when a variable has a unique\noccurrence in the formula, then the clause it appears in can always be satisfied. Hence the\nsubformula obtained by removing this clause is equivalent (in terms of satisfiability) to the\noriginal system [11]. The interest of this remark is that it can be iterated, and more and more"},{"paragraph_id":"p4","order":4,"text":"clauses eliminated. Monitoring the evolution of the formula under this procedure, called leaf\nremoval, provides us with useful information on the nature of the solution space [12, 13, 14].\nOne clause is removed at each time step. After T steps we denote by Cj(T) the number of\nclauses of length j. Those numbers obey the evolution equations (in expectation),\nCj(T + 1) −Cj(T) = −\nj Cj(T)\nPk\nj′=2 j′Cj′(T)\n(1)\nwhere the denominator is the total number of occurrences of all variables appearing in the\nformula. The r.h.s. of (1) is simply (minus) the probability that the unique-occurrence variable\nis drawn from a clause of length j.\nIn addition let us define the number Nl(T) of variables appearing in lequations exactly. The\nevolution equations for those numbers are (in expectation)\nNl(T + 1) −Nl(T) =\nk\nX\nj=2\nj(j −1) Cj(T)\nPk\nj′=2 j′Cj′(T)\n×\n (l+ 1) Nl+1(T) −lNl(T)\nP∞\nl′=0 l′Nl′(T)"},{"paragraph_id":"p5","order":5,"text":"−δl,1 + δl,0 .\n(2)\nThe above is easy to interpret. The second term in the square bracket on the r.h.s. is the\naverage number of removed variables (other than the single-occurrence variable), that is, the\naverage length of the removed clause minus one. The first term expresses that, if one of those\nvariables appeared l+ 1 times before its removal, the number of its occurrences has decreased\ndown to lafter the removal. Finally, the two δ correspond to the elimination from the system\nof the single-occurrence variable.\nIn the large N limit we may turn those finite difference equations over extensive quantities\nCj, Nlinto differential equations for their intensive counterparts cj = Cj/N, nl= Nl/N as\nfunctions of the reduced number of steps, τ = T/N. The outcome is\ndcj\ndτ\n=\n−jcj\nN ,\n(j = 2, . . . , k) ,\n(3)\ndnl\ndτ\n=\nk\nX\nj=2\nj(j −1)cj\nN\n (l+ 1)nl+1 −lnl\nN"},{"paragraph_id":"p6","order":6,"text":"−δl,1 + δl,0 ,\n(4)\nwhere N(τ) = Pk\nj=2 jcj(τ) = P\nl≥1 lnl(τ). The initial conditions are\ncj(0) = C0\nj\nN ;\nnl(0) = e−λ0 (λ0)l\nl!\n,\n(5)\nwhere λ0 is determined by P\nllnl(0) = λ0 = P\nj jcj(0).\nIt is easy to check that equations (3) are solved by cj(τ) = cj(0) b(τ)j provided N\nb\ndb\ndτ = −1.\nIt is convenient to introduce the generating function\nG(b) =\nk\nX\nj=2\ncj(0) bj .\n(6)\nDerivative(s) of G with respect to its argument will be denoted by prime(s). We have that\nN(τ) = b(τ)G′(b(τ)).\nIn addition, we define γ(τ) = P\nj cj(τ) = G(b(τ)).\nWe deduce the\nequation for b(τ):\ndγ\ndτ = N\nb\ndb\ndτ = −1\n⇒\nτ = γ(0) −γ(τ) =\nk\nX\nj=2\ncj(0)(1 −b(τ)j) .\n(7)"},{"paragraph_id":"p7","order":7,"text":"The interpretation of the equation above is just that at each step of the leaf removal one equation\nis eliminated.\nThe solution to (4) remains Poissonian at all times for all l≥2.\nSubstituting nl(τ) =\ne−λ(τ) λ(τ)l\nl!\nwe obtain an equation for λ(τ):\ndλ\ndτ = −\nP\nj≥2 j(j −1)cj(τ)\n(P\nj≥2 jcj(τ))2\nλ(τ) = −\n G′′(b)\nG′(b)2"},{"paragraph_id":"p8","order":8,"text":"b=b(τ)\nλ(τ) ,\n(8)\nwith the initial condition imposed by λ(0) = λ0 = P\nj jcj(0) = G′(1).\nFrom (7) we get\ndτ\ndb = −G′(b) so that\ndλ\ndb = dλ\ndτ\ndτ\ndb = G′′(b)\nG′(b) λ ,\n(9)\nwhich is solved by\nλ(b) = G′(b) ,\n(10)\nwhere the normalization is fixed by the initial condition for λ. (7) and (10) determine b(τ) and\nλ(τ), which describe the evolution of the formula under the action of the leaf removal algorithm.\n3.2. Static Phase Transitions\nThe structure of the subformula remaining at the end of the leaf-removal (if any) is indicative\nof the nature of the phase corresponding to typical formulas, uniformly drawn at fixed {C0\nj }.\nThree phases are possible: the unclustered phase where formulas are satisfiable and the solutions\nform a unique cluster; the clustered phase where solutions are divided into many clusters; and\nthe unsat phase where the typical formula is not satisfiable\n(i) Clustering transition:\nThe leaf removal algorithm starts from b = 1, then b decreases\naccording to (7) and the algorithm stops at the largest value of b such that n1 = 0, i.e.\nthere are no more variables with unique occurrence. We have\nn1\n=\nk\nX\nj=2\njcj −\nX\nl>1\nlnl= bG′(b) −\nX\nl>1\nle−λ(b) λ(b)l\nl!\n=\nbλ(b) −e−λ(b)λ(b)\nh\neλ(b) −1\ni\n= λ(b)\nh\nb −1 + e−λ(b)i\n,\ntherefore\nn1 = 0\n⇔\n1 −b = e−λ(b) = e−G′(b) .\n(11)\nThis equation always has the solution b = 0, that gives cj = 0 for all j when the algorithm\nstops. This corresponds to a backbone-free formula whose solution space is connected. On\nthe other hand, if this equation admits non-trivial solutions b > 0, the algorithm stops when\nb is equal to the largest of them, i.e. it is unable to eliminate all clauses in the formula.\nThen the space is clustered and the largest solution represents the fraction of variables in\nthe backbone of each cluster [12, 13].\nIn the pure (k, d)-UE-CSP case, i.e. when c0\nj = αδj,k, the critical ratio at which clustering\nappears decreases with k, from αd(3) ≃0.818 to αd(k) ≃log k/k at large k.\n(ii) Sat/unsat transition: The formula is satisfiable when the subformula left by the removal\nalgorithm has a solution. This happens with high probability if and only if the number of\nequations, given by G(b), is smaller than the number of variables, P\nl≥2 nl[12, 13]. Using\nthe condition n1 = 0, the satisfiability condition is\nG(b) ≤b + (1 −b) log(1 −b) .\n(12)\nFor (k, d)-UE-CSP, the critical ratio at which formulas go from typically satisfiable to\ntypically unsatisfiable increases with k, from αs(3) ≃0.918 to αd(k) →1 at large k."},{"paragraph_id":"p9","order":9,"text":"3.3. The potential for the backbone\nThe outcome of the previous section can be summarized as follows. We considered a formula\nspecified by a set {c0\nj}j=2,···,k, or equivalently by the generating function (6). In the following\nwe will drop the superscript 0 to simplify the notation. We define the potential\nV (b) = −G(b) + b + (1 −b) log(1 −b) .\n(13)\nThe condition n1 = 0 (11), is equivalent to V ′(b) = 0. Thus, if V (b) has a single minimum in\nb = 0, the solution space is not clustered, while if there is another minimum at b ̸= 0, there\nare clusters. Moreover, the condition for satisfiability (12), is that at the secondary minimum\nV (b) ≥0. Examples are given in figure 2.\nThe sat/unsat surface Σs, that separates the sat and the unsat phase, is defined by the\ncondition:\nΣs ≡{cj : V (b) = 0 and V ′(b) = 0 admit a solution b > 0} .\n(14)\nThe clustering surface Σd, that separates the clustered and unclustered regions, is defined\nsimilarly by\nΣd ≡{cj : V ′(b) = 0 and V ′′(b) = 0 admit a solution b > 0} .\n(15)\nThe equations above have to be interpreted as coupled equations for (b, cj); therefore Σs, Σd\nhave dimension k −2 and are surfaces in the space {cj}j=2,···,k of dimension k −1. Note that\nin (14) and (15), one must always choose the largest solution for b, to which we will refer as bs\nand bd, respectively.\nIn addition to the previous sets, in the following a special role will be played by the condition\n2c2 = 1, or equivalently V ′′(0) = 0, that defines the contradiction surface Σq:\nΣq ≡{cj : V ′′(0) = 0} .\n(16)\nThe surface Σq is simply a hyperplane of dimension k −2.\n3.4. The phase diagram\nWe draw a phase diagram in the space of the cj by representing surfaces Σs, Σd, Σq. We focus\non the region cj ∈[0, 1] for j = 3, . . . , k and c2 ∈[0, 1/2]. Indeed, if one of the cj > 1, the\nsystem is surely in the unsat phase [7] while if c2 > 1/2 the algorithm discussed above find a\ncontradiction with very high probability.\nExamples of the phase diagram are in figure 1 for k = 3 and k = 4. There are some special\n“lines” (i.e. intersections of surfaces) on which we will concentrate.\n(i) Recall that Σq is defined by V ′′(0) = 0 and note that V ′(0) = 0 for all b, cj. Thus, on\nΣq, the point b = 0 is a solution of both equations (14) and (15). The surfaces Σs, Σd are\ndefined by the existence of solutions with b > 0, but they might intersect Σq if for some\nvalues of {cj} the solution with b > 0 merges with the solution b = 0. This happen when\nV ′′′(0) = 0, as this is the limiting case in which a saddle at b = bd > 0 and a secondary\nminimum at b = bs > 0 can merge for bd, bs →0. The condition V ′′′(0) = 0 is equivalent to\nc3 = 1/6, and this defines the k −3-dimensional surface\nΣcrit ≡{cj : c2 = 1/2, c3 = 1/6} ,\n(17)\nto which we will refer as critical surface. It is easy to see that the three surfaces Σs, Σd, Σq\nare tangent to each other on the region of the critical surface where they intersect. To\nshow that one must consider a displacement c3 = 1/6 + ε and show that (15), (14) admit a\nsolution with bs, bd ∼ε if c2 −1/2 ∼ε2. We say that in this case the phase transitions are\nof second order because the order parameter b vanishes continuously at the transition."},{"paragraph_id":"p10","order":10,"text":"c\nc\nc2\n3\n4\n1/2\n1/6\n0\n0.2\n0.4\n0.6\n0.8\n1\nc3\n0.2\n0.4\n0.6\n0.8\n1\nc4\n0\n0.1\n0.2\n0.3\n0.4\n0.5\nc2\n0.2\n0.4\n0.6\n0.8\n1\nc3\nFigure 1. (Left) Schematic phase diagram of k=4-UE-CSP. The full (black) curve is the surface\nΣd, the dot-dashed (red) surface is Σs. The two surfaces meet along a portion of the line Σcrit,\ndefined by c2 = 1/2 and c3 = 1/6 and represented as a dashed (blue) line. (Right, top and\nbottom) The sections of Σd (full, black) and of Σs (dot-dashed, red), at fixed c2 (= 0, 0.1, 0.2,\n0.3, 0.4, 0.5 from top to bottom) as a function of c3 on the top panel, and at fixed c4 (= 0, 0.1,\n0.2, 0.3, 0.4, 0.5, 0.6, 0.7 from top to bottom) as a function of c2 in the bottom one. The lines\ncorresponding to c4 = 0 also represent the phase diagram of 3-UE-CSP.\n(ii) There is no a priori reason for which the three surfaces must cross at Σcrit. In fact, the\nsolutions at b > 0 might also disappear discontinuously, like in figure 2, and the surfaces Σs\nand Σd can intersect the surface Σq in regions different from Σcrit. This does not happen\nfor k = 3 but happens for k = 4 for large c4, see figure 1. In this case the transition is called\nfirst order because the order parameter jumps at the transition.\nThe generic phase diagram for all k has the shape of the one for k = 4 which we report in\nfigure 1, left panel.\n4. Search Trajectories in the Space of Formulas\nThe heuristics we defined in section 2 enjoy the property that, after any number of steps of\nthe algorithm, the reduced formula is uniformly distributed over the set of remaining N −T\nvariables conditioned to the numbers Cj(T) of clauses of length j (= 2, ..., k) [8, 9].\nThis\nstatistical property, combined with the concentration phenomenon taking place in the large N\nlimit, allows us to study the evolution of the average clauses densities cj(t) = Cj(T)/N on the\ntime scale t = T/N (fraction of assigned variables), which defines a trajectory in the cj’s space.\nNote that these cj(t) are defined with respect to N, therefore the actual clause density for the\nreduced system of N −T variables are ecj(t) = cj(t)/(1 −t). The trajectory of the ecj(t) moves\nin the cj space of the previous section1.\nInitially we have cj(0) = α δjk, i.e. the evolution starts on the ck axis at ck = α. The\n1 The reader should keep in mind this change of notation to avoid confusion in the following arguments"},{"paragraph_id":"p11","order":11,"text":"evolution equation for the densities take the form of first order differential equations,\n ̇cj = (j + 1)cj+1 −jcj\n1 −t\n−ρj(t) .\n(18)\nThe interpretation of the equations above is the following. Let us consider an interval [t, t + dt]\nof continuous time that corresponds to ∆T ∼Ndt time steps of the algorithm. The first term\non the r.h.s. arises from the decrease by one of the length of the clauses that contained the\nvariable just assigned by the algorithms during this interval. The second term corresponds to\nan additional loss of clauses which is present when the variable is selected from a clause of\nlength j: as the heuristics explicitly chooses an equation (and a variable therein) of length j\nwith probability pj (see section 2), this equation will be reduced irrespectively of the number of\nother occurrences of the variable. Hence ρj(t) is given, for j ≥1, by\nρj(t) =\nlim\n∆T→∞lim\nN→∞\n1\n∆T\ntN+∆T−1\nX\nT=tN\n(pj −pj+1) ≡⟨pj −pj+1⟩,\n(19)\nwhere both pj, pj+1 depend on their arguments (numbers of clauses) and ⟨•⟩represents the\naverage over ∆T defined in (19). Here pk+1 ≡0. Note that the case j = 1 is special as all\nclauses of length one that are produced are immediately eliminated. On average\nρ1 ≡2c2\n1 −t\n(20)\nclauses of length 2 become of length 1 and are then eliminated by unit propagation. The total\nfraction of eliminated clauses is\n ̇γ(t) ≡−\nk\nX\nj=2\n ̇cj(t) = 2c2(t)\n1 −t +\nk\nX\nj=2\nρj(t) =\nk\nX\nj=1\nρj(t) ≤1 ,\n(21)\nwhere the last inequality follows from (19). As only clauses of length one are eliminated, the\nviolation of (21) can only happen if too many such clauses are generated. This corresponds\nto ρ1 →1−; in this case a contradiction occurs with high probability and the algorithm stops\nwith the ‘Don’t know’ output. When ρ1 →1−, the algorithm makes only unit propagations and\nρj →0+ for all j ≥2. For this reason we called the plane ρ1 = 1, i.e. ec2 = 1/2, contradiction\nsurface.\n4.1. Unit Clause (UC)\nIn the UC heuristic variables are chosen at random when there is no unit clause. Hence ρj = 0\nfor j = 2, · · · , k. The solution to (18) is cj(t) = α\n k\nj\n (1 −t)jtk−j. The algorithm will generate\na contradiction with high probability (w.h.p.) if the average number of unit clauses starts to\nbuild-up, i.e. if 2c2(t)/(1 −t) ≥1. This gives an equation for the value of α at which the\nprobability that the algorithm finds a solution vanishes: for k = 3, α(UC)\na\n= 2/3.\n4.2. Generalized Unit Clause (GUC)\nIn the GUC heuristic the algorithm always fixes a variable appearing in the shortest clauses. In\nthe continuum limit cj = 0 for j smaller than a given value; therefore we define\nj∗(t) = min{j : cj(t) > 0} ,\n(22)"},{"paragraph_id":"p12","order":12,"text":"the minimal length of clauses with positive densities. We also define\nt∗(j) = min[t : cj−1(t) > 0]\n(23)\nthe time at which j∗jumps down from j to j −1. Essentially, the algorithm picks one clause of\nlength j∗and assigns successively all the variables in this clause until the clause disappears. But\nin doing so, other clauses of length j < j∗are generated and have to be eliminated to recover\nthe situation in which Cj = 0 for all j < j∗; for this reason ρj∗is not given exactly by 1/j∗.\nWhen the number of generated clauses is so high that the algorithm is unable to remove them,\ncj∗−1 becomes different from 0 and j∗jumps down by 1. The resulting motion equations for the\nclause densities are, for j ≥j∗(t):\n ̇cj(t) = (j + 1)cj+1(t) −jcj(t)\n1 −t\n−δj,j∗(t)\n 1\nj −(j −1)cj(t)\n1 −t"},{"paragraph_id":"p13","order":13,"text":".\n(24)\nThe transition times t∗are given by\ncj(t∗(j))\n1 −t\n=\n1\nj(j −1) ,\n(25)\nwhere the algorithm is no more able to remove the clauses of length j∗because too many clauses\nof length j∗−1 are being generated by propagations.\nComparing with (18) above, we observe that in the interval t ∈[t∗(j+1), t∗(j)], where j∗= j,\nonly two ρj are different from 0:\nρj∗= 1\nj∗−(j∗−1)cj∗(t)\n1 −t\n,\nρj∗−1 = j∗cj∗(t)\n1 −t\n,\n(26)\nthe first representing clauses of length j∗which are directly eliminated, the second representing\nthe clauses of length j∗−1 that are produced and subsequently eliminated in the process. In\nthis interval of time, the ratio cj∗(t)/(1 −t) increases from 0 to 1/j∗/(j∗−1) from condition\n(25). Then\n1\nj∗(t) ≤ ̇γ(t) = (ρj∗+ ρj∗−1) ≤\n1\nj∗(t) −1 ,\n(27)\nwhich is consistent with (but stronger than) (21) above.\n5. Analysis of the “dynamic” phase diagram\nConsider now a given heuristic, and a generic (k, d)-UE-CSP formula specified by its clause-to-\nvariable ratio α. The formula, in the cj space, starts on the axis ck at ck = α. The evolution\nof the formula under the action of the algorithm is represented by a trajectory {cj(t, α)}j=2,···,k\nor equivalently by G(b; t, α) = Pk\nj=2 bjcj(t, α), that depends on α through the initial condition\nG(b; 0, α) = αbk. We define a potential V (b; t, α) by replacing in (13) G(b) →G(b; t, α)/(1 −t);\nthe normalization (1 −t) is due to the fact that the cj = Cj/N are divided by N instead of\nN −T.\nWe follow the evolution of the formula by looking at the times at which the trajectory\nstarting at ck = α at time 0 crosses the surfaces Σs, Σd, Σq defined in section 3.3, which we call\nts(α), td(α), tq(α) respectively. As an example, in figure 2 we report the potential at different\ntimes during the evolution of a formula according to the UC heuristic for α > α(UC)\na\n.\nWe draw a “dynamic phase diagram” by representing in the (t, α) plane the lines separating\nthe unclustered, clustered, unsat and contradiction phases, which we call αd(t), αs(t), αq(t) and"},{"paragraph_id":"p14","order":14,"text":"0\n0.2\n0.4\n0.6\n0.8\n1\nb\n0\n0.05\n0.1\n0.15\n0.2\nV(b)\nFigure 2.\nAn example of the potential V (b; t, α) plotted (from top to bottom) at times\nt = {0, td = 0.02957, 0.07327, ts = 0.11697, 0.20642} during the evolution of a (3, d)-UE-CSP\nformula with α = 0.8 under the UC heuristic. In the unclustered region it is a convex function\nof b with a global minimum in b = 0. On the clustering line td it first develops a secondary\nminimum. On the sat/unsat line the value of V at the secondary minimum becomes equal to 0.\nare just the inverse of the times defined above.\nExamples in the case of the UC and GUC\nheuristics are given in figure 3.\nFrom the general properties of the function V (b; t, α) we can deduce a number of properties\nof the lines αd(t), αs(t), αq(t). We will show that the three lines intersect at a “critical point”\n(ta, αa), located at αa ≤αd, under the more general conditions. This implies that the algorithm\nstops working at the value αa ≤αd, which is our central result: Poissonian search algorithm\ncannot find a solution in polynomial time in the clustered region.\n5.1. Equations for the transition lines\nThe generating function G(b; t, α) satisfy an evolution equation which is easily derived from (18):\n ̇G(b; t, α)\n=\n1 −b\n1 −tG′(b; t, α) −F(b; t, α) ,\n(28)\nF(b; t, α)\n≡\n2c2(t)\n1 −t b +\nk\nX\nj=2\nρj(t)bj =\nk\nX\nj=1\nρj(t)bj .\n(29)\nPerforming the total derivative with respect to t of the first condition (V ′ = 0) in (15) for\n(αd, bd) and using the second condition, V ′′ = 0, we have\n∂V ′\n∂α\ndαd\ndt + ̇V ′ = 0\n⇒\ndαd\ndt = −\n ̇V ′(bd; t, αd)\n∂V ′\n∂α (bd; t, αd) .\n(30)\nUsing the definition (13) we have\n ̇V ′(b; t, α)\n=\n−\n1\n1 −t"},{"paragraph_id":"p15","order":15,"text":"̇G′(b; t, α) + G′(b; t, α)\n1 −t"},{"paragraph_id":"p16","order":16,"text":",\n(31)"},{"paragraph_id":"p17","order":17,"text":"0.6\n0.7\n0.8\n0.9\n1\nα\n0\n0.2\n0.4\n0.6\nt\nαd\nαs\nαq\nUC\nGUC\nα\nt\nFigure 3. (Left) Phase boundary lines in the (t, α) plane for the UC and GUC heuristics for\nk = 3. The three lines meet at the critical point (ta, αa) at which the algorithm is no more\nable to find a solution (black dot). (Right) The generic shape of the clustering and of the\nsat/unsat lines. The possibility of a maximum cannot be excluded, but in any case t must be a\nsingle-valued function of α, meaning that if the algorithm enters the cluster (or unsat) phase it\ncannot escape at later times.\n∂V ′\n∂α\n=\n−\n1\n1 −t\n∂G′\n∂α = −\n1\n1 −t\nX\nj≥2\njbj−1 ∂cj(t, α)\n∂α\n.\n(32)\nThen\ndαd\ndt = −\n ̇G′(b; t, α) + G′(b;t,α)\n1−t\n∂αG′(b; t, α)"},{"paragraph_id":"p18","order":18,"text":"α=αd(t),b=bd(t)\n.\n(33)\nUsing (28) and differentiating with respect to b we have\n ̇G′(b; t, α) + G′(b; t, α)\n1 −t\n= 1 −b\n1 −tG′′(b; t, α) −F ′(b; t, α) .\n(34)\nNow using V ′′(b; t, α) = −G′′(b;t,α)\n1−t\n+\n1\n1−b and V ′′(bd, t) = 0 we have 1−b\n1−tG′′(b; t, α) = 1 for b = bd\nand finally we get\ndαd\ndt = −1 −F ′(b; t, α)\n∂αG′(b; t, α)"},{"paragraph_id":"p19","order":19,"text":"α=αd(t),b=bd(t)\n.\n(35)\nA very similar reasoning leads to the following equation for the sat/unsat line:\ndαs\ndt = −b −F(b; t, α)\n∂αG(b; t, α)"},{"paragraph_id":"p20","order":20,"text":"α=αs(t),b=bs(t)\n.\n(36)\nThe equation for the contradiction line is easily derived from its definition ec2(t, α) = c2(t,α)\n1−t\n= 1\n2,\nwhich immediately gives\ndαq\ndt = −1 + 2 ̇c2(t, α)\n2∂αc2(t, α)"},{"paragraph_id":"p21","order":21,"text":"α=αq(t)\n.\n(37)"},{"paragraph_id":"p22","order":22,"text":"5.2. General properties of the transition lines\nWe wish to show that the transition lines td(α),ts(α) and tq(α) in the (α, t) plane are single-\nvalued functions of α, and that they meet in a point (αa, ta) where they have infinite slope and\nare therefore tangent to each other; the value αa correspond to a trajectory which is tangent to\nthe crytical surface Σcrit.\nOur argument goes as follows:\n(i) We defined αa as the value of α for which the probability of finding a solution for the\nchosen heuristic vanishes. Then the trajectory2 corresponding to any α > αa must cross\nthe contradiction surface, while the trajectory corresponding to any α < αa must not cross\nit, so that the trajectory corresponding to αa must be tangent to the contradiction surface\nΣq. The latter trajectory is tangent to Σq when ec2(t) = 1/2,\nd\ndtec2(t) = 0; the solution to\nthese conditions gives ta and αa.\nMoreover, ec2(t) = 1/2 implies that ρ1 = 1 which then implies ρj = 0 for all j ≥2, as already\ndiscussed. Then we have\nd\ndtec2(t) = d\ndt\n2c2(t)\n1 −t = 2 ̇c2(t)\n1 −t + 2c2(t)\n(1 −t)2 = 0\n⇒\n ̇c2(t) = −c2(t)\n1 −t = −1\n2 ,\n(38)\nwhich, together with the equations of motion (18) and ρ2 = 0 gives\n−c2(t)\n1 −t = dc2(t)\ndt\n= 3c3(t) −2c2(t)\n1 −t\n⇒\nec3(t) = c3(t)\n1 −t = 1\n3\nc2(t)\n1 −t = 1\n6 .\n(39)\nTherefore the point where the trajectory for α = αa is tangent to the contradiction surface\nbelongs to the critical surface Σcrit. From equation (37) it is clear that since ̇c2 = −1/2,\nthe function tq(α) has infinite slope in (ta, αa), as in figure 3.\n(ii) Next we show that the numerators of the fractions appearing in ̇αd(t) and ̇αs(t) are strictly\npositive if t < tq(α), i.e. in before a contradiction is found. Using the definition (19) we\ncan write:\nF(b; t, α)\n=\nk\nX\nj=1\nρj(t)bj = b"},{"paragraph_id":"p23","order":23,"text":"⟨p1⟩+\nk\nX\nj=2\nbj−2(b −1) ⟨pj⟩"},{"paragraph_id":"p24","order":24,"text":",\n(40)\nF ′(b; t, α)\n=\nk\nX\nj=1\njρj(t)bj−1 = ⟨p1⟩+\nk\nX\nj=2\nbj−2 [1 −j(1 −b)] ⟨pj⟩.\nThe coefficients in front of ⟨pj⟩≥0 in the sums above are always smaller than 1,\nindependently of j, so that\nF(b; t, α)\n≤\nb"},{"paragraph_id":"p25","order":25,"text":"⟨p1⟩+\nk\nX\nj=2\n⟨pj⟩"},{"paragraph_id":"p26","order":26,"text":"≤b ,\n(41)\nF ′(b; t, α)\n≤\n⟨p1⟩+\nk\nX\nj=2\n⟨pj⟩≤1 .\n(42)\nThe functions F(b; t, α) and F ′(b; t, α) are to be computed in b = bs(t, α) or b = bd(t, α)\nin equations (35) and (36). Both bs and bd are strictly smaller than 1 for all (t, α), as one\ncan directly show from their definitions because V ′(b →1) →∞. Then the coefficients in\nthe sums in (40) are strictly smaller than 1, and the only solution to F = b or F ′ = 1 is\n⟨pj⟩= δ1j, which happens only on the contradiction line.\n2 Recall that we are here talking about average trajectories."},{"paragraph_id":"p27","order":27,"text":"(iii) The denominators in equations (35), (36) are surely positive at t = 0, as G(b; 0, α) = αbk\nindependently of the heuristic. If they remain positive at all times, then ̇αd(t), ̇αs(t) ≤0 at\nall times, or equivalently dtd\ndα , dts\ndα ≤0 at all α, so that td, ts always increase on decreasing α.\nThe other possibility is that the denominator in (35) crosses zero and become negative,\nleading to a maximum in td(α), which will then decrease on decreasing α. Possibly the\ndenominators can vanish again, giving rise to a sequence of maxima and minima, see right\npanel of figure 3.\nWhat is important is that the numerator is always strictly positive, and as a consequence\ntd(α) or ts(α) are single-valued functions of α. In fact, for td(α) or ts(α) to be multiple-\nvalued functions of α, at some point their slope must become infinite, which is excluded by\nthe analysis above.\n(iv) The statement above, that td(α) and ts(α) are single valued functions of α, implies that\nif a trajectory enters the clustered or unsat phase, it cannot exit from it. This is enough\nto show that αa ≤αd; in fact, the trajectory for α = αa cannot start inside the clustered\nphase, as it would not be able to escape and reach the origin, which is required to find a\nsolution.\n(v) In general the function ec2(t) increases until it reaches a maximum and then decreases\nto 0.\nFor α = αa the value at the maximum is ec2 = 1/2.\nFor α > αa, the value\nat the maximum is ec2 > 1/2, therefore the contradiction ec2 = 1/2 is reached before\nthe maximum, when ec2 is still increasing.\nThen\nd\ndtec2 > 0 at the contradiction point.\nPerforming a simple computation similar to equations (38), (39), one can show that the\ntrajectories for α > αa meet the contradiction surface at ec3 > 1/6.\nNotice then that,\nas it is evident in figure 1, the trajectories corresponding to α > αa must enter first the\nclustered and then the unsat phases in order to reach the contradiction surface, therefore\nfor α < αa one has td(α) < ts(α) < tq(α). On the contrary the trajectories corresponding\nto α < αa must stay away from the clustering and sat/unsat surfaces, otherwise they could\nnot exit and should meet the contradiction surface: therefore for any α < αa, td(α) and\nts(α) do not exist. For α →α+\na , as the surfaces Σd, Σs, Σq are tangent in Σcrit, one has\ntd(αa) = ts(αa) = tq(αa) = ta and the three curves have infinite slope as all the numerators\nin equations (35), (36), (37) vanish on the contradiction surface. This is indeed what is\nobserved in figure 3 for the UC and GUC heuristics, and this argument confirms that this\nis the generic behavior for all the heuristics in the class considered here.\nThis structure is particularly evident for UC, where\nG(UC)(b; t, α) = α[1 −(1 −b)(1 −t)]k −αtk−1[kb(1 −t) + t] .\n(43)\nFrom (43) it is straightforward to check that ∂αG(b; t, α) > 0, ∂αG′(b; t, α) > 0, if b > 0. Then,\nas F(b; t, α) = 2bc2(t)\n1−t\nfor UC, both ̇αd(t) and ̇αs(t) are proportional to 2c2(t)\n1−t −1. This means\nthat αs,αd are decreasing functions of t below the contradiction line.\nThe conclusion is that for a generic Poissonian heuristic, the three lines cross at a critical\npoint (ta, αa) which depends on the heuristic. Above αa the heuristic will cross all the lines\nand find a contradiction. From the properties of the dynamical line, we have that generically\nαa ≤αd, that is no Poissonian search heuristic can find a solution in polynomial time above\nαd, as stated at the beginning of this section. The natural question is then if there exists an\nheuristic that saturates the bound, i.e. such that αa = αd. From the discussion above it is clear\nthat this is possible only if ̇αd(t) ≡0, i.e. the dynamical line in the (t, α) plane is a straight\nvertical line, which is possible only if the numerator in (35) is identically vanishing."},{"paragraph_id":"p28","order":28,"text":"5.3. Optimality of GUC\nIt is quite easy to see that GUC is the heuristic that locally optimizes the numerator in (35).\nIndeed, from the definition F ′(b; t, α) = Pk\nj=1 jbj−1ρj and the bound F ′(1, t) ≤1, it is clear\nthat F ′(b; t, α) is maximized by maximizing ρj for the smallest possible j, i.e. by picking clauses\nfrom the shortest possible ones, that is GUC. Unfortunately a general proof of the optimality\nof GUC for finite k seems difficult, because one should prove that GUC optimizes globally the\nclustering line, and also control the denominator in (35). In this section we will show that for\nk →∞, GUC is optimal in the sense that ̇αd ≡0 and αd = αa at leading order in k.\nFrom the definition γ(t) = −Pk\nj=2 cj(t) and integrating over time the bound (27), we have\nfor GUC:\nα −\nZ t\n0\ndt′\nj∗(t′) −1 ≤−γ(t) ≤α −\nZ t\n0\ndt′\nj∗(t′) .\n(44)\nor, equivalently,\nα −\nX\nj\nt∗(j) −t∗(j + 1)\nj −1\n≤−γ(t) ≤α −\nX\nj\nt∗(j) −t∗(j + 1)\nj\n.\n(45)\nwhere the sums are limited to the values of j that are reached during the search. In the large k\nlimit, provided the hypothesis\nt∗(j) −t∗(j + 1) = 1\nk + o(1/k)\n(46)\nholds for most j, we obtain\n−γ(t) ≃α −1\nk\nk\nX\nj∗(t)\n1\nj .\n(47)\nThe hypothesis (46) is well supported by numerical data, as shown in figure 4. As the sum of the\ninverse of the first k integers is equivalent to log k (harmonic number) we see that the minimal\nvalue of j∗over t is much larger than 2 if α is much smaller than log k/k. Therefore\nαa ≥log k\nk\n.\n(48)\nThe r.h.s. of the above inequality coincides with the asymptotic scaling of the clustering critical\nratio (section 3.2). Since the results of the previous section require that αa ≤αd, we obtain\nthat α(GUC)\na\n= αd ≃log k/k at the leading order in k →∞. As a comparison, it is easy to see\nthat for UC the threshold for large k is α(UC)\na\n≃e/k, which is therefore much lower than the\nthreshold for GUC.\nThese arguments are supported by numerical simulations that we performed up to k = 216,\nin which the equations of motion (24) are integrated as finite differences equations for all values\nof j (see figure 4). The numerical investigation confirms that kα(GUC)\na\nis very well fitted by\nlog k + 2.15 for k in the range 28 ÷ 216. Moreover, a finite size scaling analysis (with respect to\nk) of the data shown in figure 4 shows that\nk[t∗(j) −t∗(j + 1)] = 1 + kν × f(j/k)\n(49)\nwhere f(x) is a function independent on k which behaves as x−μ for x close to 0. From the\nnumerical data, it appears that ν = μ = 0.5, which confirms that the first correction to the\nleading term log k/k is of order 1/k."},{"paragraph_id":"p29","order":29,"text":"0\n0.2\n0.4\n0.6\n0.8\n1\nj k\n0.7\n0.8\n0.9\n1\n1.1\n1.2\n1.3\nk Dt*\n5\n6\n7\n8\n9\n10\n11\n12\nlogHkL\n7\n8\n9\n10\n11\n12\n13\n14\nk Αa\n0\n0.2\n0.4\n0.6\n0.8\n1\nj k\n-7.5\n-5\n-2.5\n0\n2.5\n5\n7.5\n10\n !!!k Hk Dt - 1L\n-5\n-4\n-3\n-2\n-1\nlogHxL\n0.5\n1\n1.5\n2\n2.5\n3\nlogI !!!k Hk Dt - 1LM\nFigure 4. Finite size scaling results for GUC at large k. Top Left Each curve shows the values\nof k[t∗(j) −t∗(j + 1)] as a function of j/k for k = 28, 29, . . . , 216 (from the farthest to the closest\ncurve to 1), and was obtained by integrating the equations of motion (24) by finite differences.\nFor each k, the value of α used is αGUC\na\n(k), determined as the value of α for which the maximum\nreached by 2c2(t)/(1 −t) is 1. Top Right Data points of αGUC\na\n(k) versus log k/k + 2.15/k (full\nred line). Bottom left The same data as above, plotted as {k × [t∗(j) −t∗(j + 1)]} × k1/2. The\ncurves “collapse”, showing f(x) and confirming the value of ν = 1/2. Bottom right By plotting\nthe same curves on logarithmic scale it is easily seen that for x close to 0 f(x) ≃x−μ with\nμ = 1/2, corresponding to the slope of the full red line.\n6. Conclusions\nOne of the main results of this paper, that is, that linear-time search heuristic are not able to\nsolve instances in the clustered phase of UE-CSP problems should be interpreted with care. In\nXORSAT-like models the clustering transition coincide with the emergence of strong correlations\nbetween variables in solutions, while the two phenomena generally define two distinct critical\nratios for other random decision problems [15, 16]. From an intuitive point of view it is expected\nthat the performances of search heuristics are affected by correlations between variables rather\nthan the clustering of solutions. Indeed, as the search algorithms investigated here do not allow\nfor backtracking or corrections of wrongly assigned variables, very strong correlations between\nO(N) variables (recall that the backbone includes O(N) variables in the clustered phase) are\nlikely to result in e−O(N) probabilities of success for the algorithm.\nExtending the present work to the random Satisfiability (k-SAT) problem would be interesting\nfrom this point of view, because even if the clustering and freezing transition coincide at leading"},{"paragraph_id":"p30","order":30,"text":"order for k →∞[3], their finite k values are different in this case. Moreover, in some similar\nproblems (k-COL [17] and 1-in-k-SAT [18]) it has been proven that search algorithms similar to\nthe ones investigated here are efficient beyond the point where the replica-symmetry-breaking\nsolution is stable.\nTherefore these algorithms might beat the clustering threshold in these\nproblems. Note however that in these cases the transition is continuous, so that the structure\nof the clusters is expected to be very different from the one of XORSAT.\nIn addition, while the Generalized Unit Clause heuristic is here shown to be optimal for the\nk-XORSAT problem and to saturate the clustering ratio when k →∞, it is certainly not the\ncase of the k-SAT problem. Determining a provably optimal search heuristic for this problem\nremains an open problem.\n[1] M ́ezard M, Parisi G and Virasoro M A 1987 Spin glass theory and beyond (Singapore: World Scientific)\n[2] Biroli G, Monasson R and Weigt M 2000 Eur. Phys. J. B 14 551\n[3] Krzakala F, Montanari A, Ricci-Tersenghi F, Semerjian G and Zdeborov ́a L 2007 Proc. Natl. Acad. Sci. USA\n104 10318\n[4] Montanari A and Semerjian G 2006 J. Stat. Phys. 124 103\n[5] Monasson R 2007 Introduction to Phase Transitions in Random Optimization Problems, Lecture Notes of\nthe Les Houches Summer School on Complex Systems, Elsevier\n[6] Krzakala F and Kurchan J 2007 Phys. Rev. E 76 021122\n[7] Connamacher H 2004 A Random Constraint Satisfaction Problem That Seems Hard for DPLL, Proceedings\nof the Seventh International Conference on Theory and Applications of Satisfiability Testing\n[8] Chao M T and Franco J 1990 Information Science 51 289\nChao M T and Franco J 1986 SIAM Journal on Computing 15 1106\n[9] Achlioptas D 2001 Theor. Comp. Sci. 265 159\n[10] Achlioptas D, Beame P and Molloy M 2004 J. Comput. Syst. Sci. 68 238\n[11] Dubois O and Mandler J 2002 The 3-XORSAT Threshold, Proceedings of the 43rd Symposium on Foundations\nof Computer Science\n[12] M ́ezard M, Ricci-Tersenghi F and Zecchina R, 2003 J. Stat. Phys. 111 505\n[13] Cocco S, Dubois O, Mandler J and Monasson R 2003 Phys. Rev. Lett. 90 047205\n[14] Weigt M 2002 Eur. Phys. J. B 28 369\n[15] Semerjian G 2007 On the freezing of variables in random constraint satisfaction problems Preprint\narXiv:cond-mat/07052147 (J.Stat.Phys. in press)\n[16] Krzakala F and Zdeborov ́a L 2007 Phys. Rev. E 76 031131\n[17] Achlioptas D and Moore C 2003 J. Comput. Syst. Sci. 67 441\n[18] Raymond J, Sportiello A and Zdeborov ́a L 2007 Phys. Rev. E 76 011101"}],"pages":[{"page":1,"text":"arXiv:0709.0367v2 [cs.CC] 18 Oct 2007\nProceedings of the International Workshop on Statistical-Mechanical Informatics\nSeptember 16–19, 2007, Kyoto, Japan\nRelationship between clustering and algorithmic\nphase transitions in the random k-XORSAT model\nand its NP-complete extensions\nFabrizio Altarelli1,2, R ́emi Monasson2 and Francesco Zamponi2,3\n1 Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, P.le A. Moro 2, 00185 Roma,\nItaly\n2 CNRS-Laboratoire de Physique Th ́eorique, Ecole Normale Sup ́erieure, 24 rue Lhomond,\n75005 Paris, France\n3 Service de Physique Th ́eorique, Orme des Merisiers, CEA Saclay, 91191 Gif-sur-Yvette\nCedex, France\nE-mail: fabrizio.altarelli@roma1.infn.it\nAbstract.\nWe study the performances of stochastic heuristic search algorithms on Uniquely\nExtendible Constraint Satisfaction Problems with random inputs.\nWe show that, for any\nheuristic preserving the Poissonian nature of the underlying instance, the (heuristic-dependent)\nlargest ratio αa of constraints per variables for which a search algorithm is likely to find solutions\nis smaller than the critical ratio αd above which solutions are clustered and highly correlated.\nIn addition we show that the clustering ratio can be reached when the number k of variables\nper constraints goes to infinity by the so-called Generalized Unit Clause heuristic.\n1. Introduction\nThe application of statistical mechanics ideas and tools to random optimization problems,\ninitiated in the mid-eighties [1], has benefited from a renewed interest from the discovery of\nphase transitions in Constraint Satisfaction Problems (CSP) fifteen years ago. Briefly speaking,\none wants to decide whether a set of randomly drawn constraints over a set of variables admits (at\nleast) one solution. When the number of variables goes to infinity at fixed ratio α of constraints\nper variable the answer abruptly changes from (almost surely) Yes to No when the ratio crosses\nsome critical value αs. Statistical physics studies have pointed out the existence of another\nphase transition in the Yes region [2, 3]. The set of solutions goes from being connected to a\ncollection of disconnected clusters at some ratio αd < αs, a translation in optimization terms of\nthe replica symmetry breaking transition identified by Parisi in mean-field spin glass theory.\nIt is expected that this clustering transition may have dynamical consequences. As replica\nsymmetry breaking signals a loss of ergodicity, sampling algorithms (e.g. Monte Carlo procedure)\nrun into problems at that transition. A quantitative study of the slowing down of MC scheme was\ndone in [4] for the case of the k-XORSAT model where constraints are simply linear equations\n(modulo 2) over k Boolean variables (for an introduction, see [5] and references therein). Yet,\nfinding a solution should in principle be easier than sampling, and the exact nature of the\nrelationship between the performances of resolution algorithms and the static phase transitions\ncharacterizing the solution space is far from being obvious [6]. The present paper is a modest"},{"page":2,"text":"step in elucidating this question for the k-XORSAT problem, and some related NP-complete\nproblems sharing the same random structure.\nHereafter we consider simple stochastic search heuristic algorithms working in polynomial\n(linear) time for solving k-XORSAT instances [8, 5].\nBy successively assigning variables\naccording to some heuristic rules those algorithms either produce a solution, or end up with\na contradiction. The probability that a solution is found is a decreasing function of the ratio α,\nand vanishes above some heuristic-dependent ratio αa in the infinite size limit. We show that\nαa < αd for any assignment heuristic in the class of rules preserving the Poissonian structure of\nthe instance. In addition, we determine the most efficient heuristic, that is, the one maximizing\nαa in this class and show that for large k, the two critical ratios match, αa(k) ≃αd(k) ≃log k/k.\nThe plan of the paper is as follows. In section 2 we define the random k-XORSAT decision\nproblem and its extension, as well as the search algorithms studied. Section 3 presents a method\nto characterize the phase diagrams of those random decision problems, depending on the content\n(numbers of constraints over j variables, with j ranging from 1 to k) of their instances. We\nshow that all important information is encoded in a unique ‘thermodynamical’ potential for the\nfraction of frozen variables (backbone). The analysis of the dynamical evolution of the instance\ncontent is exposed in section 4. These dynamical results are combined with the static phase\ndiagram in section 5 to show that the success-to-failure critical ratio of search heuristic, αa, is\nsmaller than the ratio corresponding to the onset of clustering and large backbones, αd. We\nthen show that the so-called Generalized Unit Clause heuristic rule is optimal (in the class of\nPoissonian heuristics) and its critical ratio αa is asymptotically equal to αd in the large k limit.\nOur results are discussed in section 6.\n2. Definitions\n2.1. Decision problems\nThe decision problems we consider in this paper are (k, d)-Uniquely Extendible (UE) Constraint\nSatisfaction Problems (CSP) defined as follows [7]. One considers N variables xi ∈{0, 1, · · · , d−\n1}. A UE constraint, or clause, is a constraint on k variables such that, if one fixes a subset of\nk −1 variables, the value of the k-th variable is uniquely determined. A (k, d)-UE-CSP formula\nis a collection of M = αN clauses, each involving k variables (out of the N available ones). A\nsolution is an assignment of the N variables such that all the clauses are satisfied. k-XORSAT\ncorresponds to d = 2 and is solvable in polynomial time with standard linear algebra techniques.\nFor d = 3 the problem is still in P, while for d ≥4 it has been shown that (3, d)-UE-CSP is\nNP-complete [7].\nA random formula is obtained by choosing, for each clause, the k variables, and the actual\nUE constraint, uniformly at random. It is known that, in the infinite size limit N →∞and at\nfixed clause-to-variable ratio α, [7, 11, 12, 13]:\n• there is a critical ratio αs(k) such that a random (k, d)-UE-CSP is almost surely satisfiable\n(respectively, unsatisfiable) if α < αs(k) (respectively, α > αs(k)).\n• in the satisfiable phase there is another phase transition at some ratio αd(k) such that:\n- for α < αd(k) the space of solutions is ‘connected’: with high probability there is\na path in the set of solutions joining any two solutions such that a step along the path\nrequires to change O(1) variables.\n- for α > αd(k) the space of solution is disconnected into an exponentially large number\nof clusters, each one enjoying the above connectedness property, and far away from each\nother (going from one solution in one cluster to another solution in another cluster requires\nto change O(N) variables). In addition, in each cluster, a finite fraction of variables are\nfrozen i.e. take the same value in all solutions (backbone)."},{"page":3,"text":"2.2. Search algorithms\nWe will consider simple algorithms acting on the formula in an attempt to find solutions. Those\nalgorithms were introduced and analyzed by Chao and Franco [8] (see [9] for a review). Briefly\nspeaking, starting from a randomly drawn formula, the algorithm assigns one variable at each\ntime step according to the following principles:\n• If there is (at least) one clause of length one (called unit-clause) then satisfy it by adequately\nassigning its variable. This rule is called unit propagation.\n• If all clauses have length two or more, then choose a variable according to some heuristic\nrules. Two simple rules are:\n- Unit Clause (UC): pick up uniformly at random any variable and set it to a random\nuniform value in {0, · · · , d −1};\n- Generalized Unit Clause (GUC): pick up uniformly at random one of the shortest\nclauses, then a variable it this clause, and finally its value.\nIn this analysis, we will discuss a general heuristics in which the variable to be set is chosen among\nthose that appear in the clauses of length j with some probability pj(C1, · · · , Ck), depending in\ngeneral on the number of clauses of length j present in the formula, that we shall call Cj. Unit\npropagation implies that if C1 ̸= 0, then pj = δj,1. We consider also the possibility that the\nvariable is chosen irrespective of the clause length, then Pk\nj=1 pj ≤1.\nBoth UC and GUC are special cases of this general class: in UC variables are chosen at\nrandom, irrespectively of the clauses they appear in (if any), so that pj = 0 unless there are\nunit clauses; GUC corresponds to pj = δj,j∗where j∗is the length of the shortest clause in the\nsystem. Notice that since the variables are selected independently of their number of occurrences,\nthe latter remains Poissonian under the action of the algorithm (even though the value of the\nparameter in the distribution of occurrences may vary). More involved heuristics do exist but\nwill not be analyzed here.\nUnder the action of the algorithm clauses get reduced (decrease in length) until they disappear\nonce satisfied. The algorithm stops either when all clauses have been satisfied or when two\nincompatible unit-clauses have been generated e.g. x = 0 and x = 1. In the latter case the\nalgorithm outputs ‘I do not know whether there is a solution’, while in the former case the\noutput reads ‘Satisfiable’ and returns a solution to the formula. The probability of success,\nthat is, the probability (over the choices of the algorithms and the formula) of getting the\n‘Satisfiable’ output vanishes above some heuristic-dependent ratio αa(< αs) in the infinite N\nlimit. This success-to-failure transition coincides with the polynomial-to-exponential transition\nof backtracking algorithms [5, 10].\n3. ‘Thermodynamical’ Characterization of the Space of Solutions\nUnder the action of the algorithm the length of the clauses changes; therefore the initial (k, d)-\nUE-CSP formula where all clauses have length k evolves into a formula with some distribution\nof clauses of different lengths. We wish then to characterize the space of solutions of a generic\nd-UE-CSP formula made by N variables and by {C0\nj }j=2,···,k clauses of length j, assuming that\nthere are no unit clauses. This characterization will be useful to analyze the performance of\nsearch algorithm in the following.\n3.1. Leaf removal procedure and its analysis\nOur starting observation is that, due to the UE property, when a variable has a unique\noccurrence in the formula, then the clause it appears in can always be satisfied. Hence the\nsubformula obtained by removing this clause is equivalent (in terms of satisfiability) to the\noriginal system [11]. The interest of this remark is that it can be iterated, and more and more"},{"page":4,"text":"clauses eliminated. Monitoring the evolution of the formula under this procedure, called leaf\nremoval, provides us with useful information on the nature of the solution space [12, 13, 14].\nOne clause is removed at each time step. After T steps we denote by Cj(T) the number of\nclauses of length j. Those numbers obey the evolution equations (in expectation),\nCj(T + 1) −Cj(T) = −\nj Cj(T)\nPk\nj′=2 j′Cj′(T)\n(1)\nwhere the denominator is the total number of occurrences of all variables appearing in the\nformula. The r.h.s. of (1) is simply (minus) the probability that the unique-occurrence variable\nis drawn from a clause of length j.\nIn addition let us define the number Nl(T) of variables appearing in lequations exactly. The\nevolution equations for those numbers are (in expectation)\nNl(T + 1) −Nl(T) =\nk\nX\nj=2\nj(j −1) Cj(T)\nPk\nj′=2 j′Cj′(T)\n×\n (l+ 1) Nl+1(T) −lNl(T)\nP∞\nl′=0 l′Nl′(T)\n \n−δl,1 + δl,0 .\n(2)\nThe above is easy to interpret. The second term in the square bracket on the r.h.s. is the\naverage number of removed variables (other than the single-occurrence variable), that is, the\naverage length of the removed clause minus one. The first term expresses that, if one of those\nvariables appeared l+ 1 times before its removal, the number of its occurrences has decreased\ndown to lafter the removal. Finally, the two δ correspond to the elimination from the system\nof the single-occurrence variable.\nIn the large N limit we may turn those finite difference equations over extensive quantities\nCj, Nlinto differential equations for their intensive counterparts cj = Cj/N, nl= Nl/N as\nfunctions of the reduced number of steps, τ = T/N. The outcome is\ndcj\ndτ\n=\n−jcj\nN ,\n(j = 2, . . . , k) ,\n(3)\ndnl\ndτ\n=\nk\nX\nj=2\nj(j −1)cj\nN\n (l+ 1)nl+1 −lnl\nN\n \n−δl,1 + δl,0 ,\n(4)\nwhere N(τ) = Pk\nj=2 jcj(τ) = P\nl≥1 lnl(τ). The initial conditions are\ncj(0) = C0\nj\nN ;\nnl(0) = e−λ0 (λ0)l\nl!\n,\n(5)\nwhere λ0 is determined by P\nllnl(0) = λ0 = P\nj jcj(0).\nIt is easy to check that equations (3) are solved by cj(τ) = cj(0) b(τ)j provided N\nb\ndb\ndτ = −1.\nIt is convenient to introduce the generating function\nG(b) =\nk\nX\nj=2\ncj(0) bj .\n(6)\nDerivative(s) of G with respect to its argument will be denoted by prime(s). We have that\nN(τ) = b(τ)G′(b(τ)).\nIn addition, we define γ(τ) = P\nj cj(τ) = G(b(τ)).\nWe deduce the\nequation for b(τ):\ndγ\ndτ = N\nb\ndb\ndτ = −1\n⇒\nτ = γ(0) −γ(τ) =\nk\nX\nj=2\ncj(0)(1 −b(τ)j) .\n(7)"},{"page":5,"text":"The interpretation of the equation above is just that at each step of the leaf removal one equation\nis eliminated.\nThe solution to (4) remains Poissonian at all times for all l≥2.\nSubstituting nl(τ) =\ne−λ(τ) λ(τ)l\nl!\nwe obtain an equation for λ(τ):\ndλ\ndτ = −\nP\nj≥2 j(j −1)cj(τ)\n(P\nj≥2 jcj(τ))2\nλ(τ) = −\n G′′(b)\nG′(b)2\n \nb=b(τ)\nλ(τ) ,\n(8)\nwith the initial condition imposed by λ(0) = λ0 = P\nj jcj(0) = G′(1).\nFrom (7) we get\ndτ\ndb = −G′(b) so that\ndλ\ndb = dλ\ndτ\ndτ\ndb = G′′(b)\nG′(b) λ ,\n(9)\nwhich is solved by\nλ(b) = G′(b) ,\n(10)\nwhere the normalization is fixed by the initial condition for λ. (7) and (10) determine b(τ) and\nλ(τ), which describe the evolution of the formula under the action of the leaf removal algorithm.\n3.2. Static Phase Transitions\nThe structure of the subformula remaining at the end of the leaf-removal (if any) is indicative\nof the nature of the phase corresponding to typical formulas, uniformly drawn at fixed {C0\nj }.\nThree phases are possible: the unclustered phase where formulas are satisfiable and the solutions\nform a unique cluster; the clustered phase where solutions are divided into many clusters; and\nthe unsat phase where the typical formula is not satisfiable\n(i) Clustering transition:\nThe leaf removal algorithm starts from b = 1, then b decreases\naccording to (7) and the algorithm stops at the largest value of b such that n1 = 0, i.e.\nthere are no more variables with unique occurrence. We have\nn1\n=\nk\nX\nj=2\njcj −\nX\nl>1\nlnl= bG′(b) −\nX\nl>1\nle−λ(b) λ(b)l\nl!\n=\nbλ(b) −e−λ(b)λ(b)\nh\neλ(b) −1\ni\n= λ(b)\nh\nb −1 + e−λ(b)i\n,\ntherefore\nn1 = 0\n⇔\n1 −b = e−λ(b) = e−G′(b) .\n(11)\nThis equation always has the solution b = 0, that gives cj = 0 for all j when the algorithm\nstops. This corresponds to a backbone-free formula whose solution space is connected. On\nthe other hand, if this equation admits non-trivial solutions b > 0, the algorithm stops when\nb is equal to the largest of them, i.e. it is unable to eliminate all clauses in the formula.\nThen the space is clustered and the largest solution represents the fraction of variables in\nthe backbone of each cluster [12, 13].\nIn the pure (k, d)-UE-CSP case, i.e. when c0\nj = αδj,k, the critical ratio at which clustering\nappears decreases with k, from αd(3) ≃0.818 to αd(k) ≃log k/k at large k.\n(ii) Sat/unsat transition: The formula is satisfiable when the subformula left by the removal\nalgorithm has a solution. This happens with high probability if and only if the number of\nequations, given by G(b), is smaller than the number of variables, P\nl≥2 nl[12, 13]. Using\nthe condition n1 = 0, the satisfiability condition is\nG(b) ≤b + (1 −b) log(1 −b) .\n(12)\nFor (k, d)-UE-CSP, the critical ratio at which formulas go from typically satisfiable to\ntypically unsatisfiable increases with k, from αs(3) ≃0.918 to αd(k) →1 at large k."},{"page":6,"text":"3.3. The potential for the backbone\nThe outcome of the previous section can be summarized as follows. We considered a formula\nspecified by a set {c0\nj}j=2,···,k, or equivalently by the generating function (6). In the following\nwe will drop the superscript 0 to simplify the notation. We define the potential\nV (b) = −G(b) + b + (1 −b) log(1 −b) .\n(13)\nThe condition n1 = 0 (11), is equivalent to V ′(b) = 0. Thus, if V (b) has a single minimum in\nb = 0, the solution space is not clustered, while if there is another minimum at b ̸= 0, there\nare clusters. Moreover, the condition for satisfiability (12), is that at the secondary minimum\nV (b) ≥0. Examples are given in figure 2.\nThe sat/unsat surface Σs, that separates the sat and the unsat phase, is defined by the\ncondition:\nΣs ≡{cj : V (b) = 0 and V ′(b) = 0 admit a solution b > 0} .\n(14)\nThe clustering surface Σd, that separates the clustered and unclustered regions, is defined\nsimilarly by\nΣd ≡{cj : V ′(b) = 0 and V ′′(b) = 0 admit a solution b > 0} .\n(15)\nThe equations above have to be interpreted as coupled equations for (b, cj); therefore Σs, Σd\nhave dimension k −2 and are surfaces in the space {cj}j=2,···,k of dimension k −1. Note that\nin (14) and (15), one must always choose the largest solution for b, to which we will refer as bs\nand bd, respectively.\nIn addition to the previous sets, in the following a special role will be played by the condition\n2c2 = 1, or equivalently V ′′(0) = 0, that defines the contradiction surface Σq:\nΣq ≡{cj : V ′′(0) = 0} .\n(16)\nThe surface Σq is simply a hyperplane of dimension k −2.\n3.4. The phase diagram\nWe draw a phase diagram in the space of the cj by representing surfaces Σs, Σd, Σq. We focus\non the region cj ∈[0, 1] for j = 3, . . . , k and c2 ∈[0, 1/2]. Indeed, if one of the cj > 1, the\nsystem is surely in the unsat phase [7] while if c2 > 1/2 the algorithm discussed above find a\ncontradiction with very high probability.\nExamples of the phase diagram are in figure 1 for k = 3 and k = 4. There are some special\n“lines” (i.e. intersections of surfaces) on which we will concentrate.\n(i) Recall that Σq is defined by V ′′(0) = 0 and note that V ′(0) = 0 for all b, cj. Thus, on\nΣq, the point b = 0 is a solution of both equations (14) and (15). The surfaces Σs, Σd are\ndefined by the existence of solutions with b > 0, but they might intersect Σq if for some\nvalues of {cj} the solution with b > 0 merges with the solution b = 0. This happen when\nV ′′′(0) = 0, as this is the limiting case in which a saddle at b = bd > 0 and a secondary\nminimum at b = bs > 0 can merge for bd, bs →0. The condition V ′′′(0) = 0 is equivalent to\nc3 = 1/6, and this defines the k −3-dimensional surface\nΣcrit ≡{cj : c2 = 1/2, c3 = 1/6} ,\n(17)\nto which we will refer as critical surface. It is easy to see that the three surfaces Σs, Σd, Σq\nare tangent to each other on the region of the critical surface where they intersect. To\nshow that one must consider a displacement c3 = 1/6 + ε and show that (15), (14) admit a\nsolution with bs, bd ∼ε if c2 −1/2 ∼ε2. We say that in this case the phase transitions are\nof second order because the order parameter b vanishes continuously at the transition."},{"page":7,"text":"c\nc\nc2\n3\n4\n1/2\n1/6\n0\n0.2\n0.4\n0.6\n0.8\n1\nc3\n0.2\n0.4\n0.6\n0.8\n1\nc4\n0\n0.1\n0.2\n0.3\n0.4\n0.5\nc2\n0.2\n0.4\n0.6\n0.8\n1\nc3\nFigure 1. (Left) Schematic phase diagram of k=4-UE-CSP. The full (black) curve is the surface\nΣd, the dot-dashed (red) surface is Σs. The two surfaces meet along a portion of the line Σcrit,\ndefined by c2 = 1/2 and c3 = 1/6 and represented as a dashed (blue) line. (Right, top and\nbottom) The sections of Σd (full, black) and of Σs (dot-dashed, red), at fixed c2 (= 0, 0.1, 0.2,\n0.3, 0.4, 0.5 from top to bottom) as a function of c3 on the top panel, and at fixed c4 (= 0, 0.1,\n0.2, 0.3, 0.4, 0.5, 0.6, 0.7 from top to bottom) as a function of c2 in the bottom one. The lines\ncorresponding to c4 = 0 also represent the phase diagram of 3-UE-CSP.\n(ii) There is no a priori reason for which the three surfaces must cross at Σcrit. In fact, the\nsolutions at b > 0 might also disappear discontinuously, like in figure 2, and the surfaces Σs\nand Σd can intersect the surface Σq in regions different from Σcrit. This does not happen\nfor k = 3 but happens for k = 4 for large c4, see figure 1. In this case the transition is called\nfirst order because the order parameter jumps at the transition.\nThe generic phase diagram for all k has the shape of the one for k = 4 which we report in\nfigure 1, left panel.\n4. Search Trajectories in the Space of Formulas\nThe heuristics we defined in section 2 enjoy the property that, after any number of steps of\nthe algorithm, the reduced formula is uniformly distributed over the set of remaining N −T\nvariables conditioned to the numbers Cj(T) of clauses of length j (= 2, ..., k) [8, 9].\nThis\nstatistical property, combined with the concentration phenomenon taking place in the large N\nlimit, allows us to study the evolution of the average clauses densities cj(t) = Cj(T)/N on the\ntime scale t = T/N (fraction of assigned variables), which defines a trajectory in the cj’s space.\nNote that these cj(t) are defined with respect to N, therefore the actual clause density for the\nreduced system of N −T variables are ecj(t) = cj(t)/(1 −t). The trajectory of the ecj(t) moves\nin the cj space of the previous section1.\nInitially we have cj(0) = α δjk, i.e. the evolution starts on the ck axis at ck = α. The\n1 The reader should keep in mind this change of notation to avoid confusion in the following arguments"},{"page":8,"text":"evolution equation for the densities take the form of first order differential equations,\n ̇cj = (j + 1)cj+1 −jcj\n1 −t\n−ρj(t) .\n(18)\nThe interpretation of the equations above is the following. Let us consider an interval [t, t + dt]\nof continuous time that corresponds to ∆T ∼Ndt time steps of the algorithm. The first term\non the r.h.s. arises from the decrease by one of the length of the clauses that contained the\nvariable just assigned by the algorithms during this interval. The second term corresponds to\nan additional loss of clauses which is present when the variable is selected from a clause of\nlength j: as the heuristics explicitly chooses an equation (and a variable therein) of length j\nwith probability pj (see section 2), this equation will be reduced irrespectively of the number of\nother occurrences of the variable. Hence ρj(t) is given, for j ≥1, by\nρj(t) =\nlim\n∆T→∞lim\nN→∞\n1\n∆T\ntN+∆T−1\nX\nT=tN\n(pj −pj+1) ≡⟨pj −pj+1⟩,\n(19)\nwhere both pj, pj+1 depend on their arguments (numbers of clauses) and ⟨•⟩represents the\naverage over ∆T defined in (19). Here pk+1 ≡0. Note that the case j = 1 is special as all\nclauses of length one that are produced are immediately eliminated. On average\nρ1 ≡2c2\n1 −t\n(20)\nclauses of length 2 become of length 1 and are then eliminated by unit propagation. The total\nfraction of eliminated clauses is\n ̇γ(t) ≡−\nk\nX\nj=2\n ̇cj(t) = 2c2(t)\n1 −t +\nk\nX\nj=2\nρj(t) =\nk\nX\nj=1\nρj(t) ≤1 ,\n(21)\nwhere the last inequality follows from (19). As only clauses of length one are eliminated, the\nviolation of (21) can only happen if too many such clauses are generated. This corresponds\nto ρ1 →1−; in this case a contradiction occurs with high probability and the algorithm stops\nwith the ‘Don’t know’ output. When ρ1 →1−, the algorithm makes only unit propagations and\nρj →0+ for all j ≥2. For this reason we called the plane ρ1 = 1, i.e. ec2 = 1/2, contradiction\nsurface.\n4.1. Unit Clause (UC)\nIn the UC heuristic variables are chosen at random when there is no unit clause. Hence ρj = 0\nfor j = 2, · · · , k. The solution to (18) is cj(t) = α\n k\nj\n (1 −t)jtk−j. The algorithm will generate\na contradiction with high probability (w.h.p.) if the average number of unit clauses starts to\nbuild-up, i.e. if 2c2(t)/(1 −t) ≥1. This gives an equation for the value of α at which the\nprobability that the algorithm finds a solution vanishes: for k = 3, α(UC)\na\n= 2/3.\n4.2. Generalized Unit Clause (GUC)\nIn the GUC heuristic the algorithm always fixes a variable appearing in the shortest clauses. In\nthe continuum limit cj = 0 for j smaller than a given value; therefore we define\nj∗(t) = min{j : cj(t) > 0} ,\n(22)"},{"page":9,"text":"the minimal length of clauses with positive densities. We also define\nt∗(j) = min[t : cj−1(t) > 0]\n(23)\nthe time at which j∗jumps down from j to j −1. Essentially, the algorithm picks one clause of\nlength j∗and assigns successively all the variables in this clause until the clause disappears. But\nin doing so, other clauses of length j < j∗are generated and have to be eliminated to recover\nthe situation in which Cj = 0 for all j < j∗; for this reason ρj∗is not given exactly by 1/j∗.\nWhen the number of generated clauses is so high that the algorithm is unable to remove them,\ncj∗−1 becomes different from 0 and j∗jumps down by 1. The resulting motion equations for the\nclause densities are, for j ≥j∗(t):\n ̇cj(t) = (j + 1)cj+1(t) −jcj(t)\n1 −t\n−δj,j∗(t)\n 1\nj −(j −1)cj(t)\n1 −t\n \n.\n(24)\nThe transition times t∗are given by\ncj(t∗(j))\n1 −t\n=\n1\nj(j −1) ,\n(25)\nwhere the algorithm is no more able to remove the clauses of length j∗because too many clauses\nof length j∗−1 are being generated by propagations.\nComparing with (18) above, we observe that in the interval t ∈[t∗(j+1), t∗(j)], where j∗= j,\nonly two ρj are different from 0:\nρj∗= 1\nj∗−(j∗−1)cj∗(t)\n1 −t\n,\nρj∗−1 = j∗cj∗(t)\n1 −t\n,\n(26)\nthe first representing clauses of length j∗which are directly eliminated, the second representing\nthe clauses of length j∗−1 that are produced and subsequently eliminated in the process. In\nthis interval of time, the ratio cj∗(t)/(1 −t) increases from 0 to 1/j∗/(j∗−1) from condition\n(25). Then\n1\nj∗(t) ≤ ̇γ(t) = (ρj∗+ ρj∗−1) ≤\n1\nj∗(t) −1 ,\n(27)\nwhich is consistent with (but stronger than) (21) above.\n5. Analysis of the “dynamic” phase diagram\nConsider now a given heuristic, and a generic (k, d)-UE-CSP formula specified by its clause-to-\nvariable ratio α. The formula, in the cj space, starts on the axis ck at ck = α. The evolution\nof the formula under the action of the algorithm is represented by a trajectory {cj(t, α)}j=2,···,k\nor equivalently by G(b; t, α) = Pk\nj=2 bjcj(t, α), that depends on α through the initial condition\nG(b; 0, α) = αbk. We define a potential V (b; t, α) by replacing in (13) G(b) →G(b; t, α)/(1 −t);\nthe normalization (1 −t) is due to the fact that the cj = Cj/N are divided by N instead of\nN −T.\nWe follow the evolution of the formula by looking at the times at which the trajectory\nstarting at ck = α at time 0 crosses the surfaces Σs, Σd, Σq defined in section 3.3, which we call\nts(α), td(α), tq(α) respectively. As an example, in figure 2 we report the potential at different\ntimes during the evolution of a formula according to the UC heuristic for α > α(UC)\na\n.\nWe draw a “dynamic phase diagram” by representing in the (t, α) plane the lines separating\nthe unclustered, clustered, unsat and contradiction phases, which we call αd(t), αs(t), αq(t) and"},{"page":10,"text":"0\n0.2\n0.4\n0.6\n0.8\n1\nb\n0\n0.05\n0.1\n0.15\n0.2\nV(b)\nFigure 2.\nAn example of the potential V (b; t, α) plotted (from top to bottom) at times\nt = {0, td = 0.02957, 0.07327, ts = 0.11697, 0.20642} during the evolution of a (3, d)-UE-CSP\nformula with α = 0.8 under the UC heuristic. In the unclustered region it is a convex function\nof b with a global minimum in b = 0. On the clustering line td it first develops a secondary\nminimum. On the sat/unsat line the value of V at the secondary minimum becomes equal to 0.\nare just the inverse of the times defined above.\nExamples in the case of the UC and GUC\nheuristics are given in figure 3.\nFrom the general properties of the function V (b; t, α) we can deduce a number of properties\nof the lines αd(t), αs(t), αq(t). We will show that the three lines intersect at a “critical point”\n(ta, αa), located at αa ≤αd, under the more general conditions. This implies that the algorithm\nstops working at the value αa ≤αd, which is our central result: Poissonian search algorithm\ncannot find a solution in polynomial time in the clustered region.\n5.1. Equations for the transition lines\nThe generating function G(b; t, α) satisfy an evolution equation which is easily derived from (18):\n ̇G(b; t, α)\n=\n1 −b\n1 −tG′(b; t, α) −F(b; t, α) ,\n(28)\nF(b; t, α)\n≡\n2c2(t)\n1 −t b +\nk\nX\nj=2\nρj(t)bj =\nk\nX\nj=1\nρj(t)bj .\n(29)\nPerforming the total derivative with respect to t of the first condition (V ′ = 0) in (15) for\n(αd, bd) and using the second condition, V ′′ = 0, we have\n∂V ′\n∂α\ndαd\ndt + ̇V ′ = 0\n⇒\ndαd\ndt = −\n ̇V ′(bd; t, αd)\n∂V ′\n∂α (bd; t, αd) .\n(30)\nUsing the definition (13) we have\n ̇V ′(b; t, α)\n=\n−\n1\n1 −t\n \n ̇G′(b; t, α) + G′(b; t, α)\n1 −t\n \n,\n(31)"},{"page":11,"text":"0.6\n0.7\n0.8\n0.9\n1\nα\n0\n0.2\n0.4\n0.6\nt\nαd\nαs\nαq\nUC\nGUC\nα\nt\nFigure 3. (Left) Phase boundary lines in the (t, α) plane for the UC and GUC heuristics for\nk = 3. The three lines meet at the critical point (ta, αa) at which the algorithm is no more\nable to find a solution (black dot). (Right) The generic shape of the clustering and of the\nsat/unsat lines. The possibility of a maximum cannot be excluded, but in any case t must be a\nsingle-valued function of α, meaning that if the algorithm enters the cluster (or unsat) phase it\ncannot escape at later times.\n∂V ′\n∂α\n=\n−\n1\n1 −t\n∂G′\n∂α = −\n1\n1 −t\nX\nj≥2\njbj−1 ∂cj(t, α)\n∂α\n.\n(32)\nThen\ndαd\ndt = −\n ̇G′(b; t, α) + G′(b;t,α)\n1−t\n∂αG′(b; t, α)\n \nα=αd(t),b=bd(t)\n.\n(33)\nUsing (28) and differentiating with respect to b we have\n ̇G′(b; t, α) + G′(b; t, α)\n1 −t\n= 1 −b\n1 −tG′′(b; t, α) −F ′(b; t, α) .\n(34)\nNow using V ′′(b; t, α) = −G′′(b;t,α)\n1−t\n+\n1\n1−b and V ′′(bd, t) = 0 we have 1−b\n1−tG′′(b; t, α) = 1 for b = bd\nand finally we get\ndαd\ndt = −1 −F ′(b; t, α)\n∂αG′(b; t, α)\n \nα=αd(t),b=bd(t)\n.\n(35)\nA very similar reasoning leads to the following equation for the sat/unsat line:\ndαs\ndt = −b −F(b; t, α)\n∂αG(b; t, α)\n \nα=αs(t),b=bs(t)\n.\n(36)\nThe equation for the contradiction line is easily derived from its definition ec2(t, α) = c2(t,α)\n1−t\n= 1\n2,\nwhich immediately gives\ndαq\ndt = −1 + 2 ̇c2(t, α)\n2∂αc2(t, α)\n \nα=αq(t)\n.\n(37)"},{"page":12,"text":"5.2. General properties of the transition lines\nWe wish to show that the transition lines td(α),ts(α) and tq(α) in the (α, t) plane are single-\nvalued functions of α, and that they meet in a point (αa, ta) where they have infinite slope and\nare therefore tangent to each other; the value αa correspond to a trajectory which is tangent to\nthe crytical surface Σcrit.\nOur argument goes as follows:\n(i) We defined αa as the value of α for which the probability of finding a solution for the\nchosen heuristic vanishes. Then the trajectory2 corresponding to any α > αa must cross\nthe contradiction surface, while the trajectory corresponding to any α < αa must not cross\nit, so that the trajectory corresponding to αa must be tangent to the contradiction surface\nΣq. The latter trajectory is tangent to Σq when ec2(t) = 1/2,\nd\ndtec2(t) = 0; the solution to\nthese conditions gives ta and αa.\nMoreover, ec2(t) = 1/2 implies that ρ1 = 1 which then implies ρj = 0 for all j ≥2, as already\ndiscussed. Then we have\nd\ndtec2(t) = d\ndt\n2c2(t)\n1 −t = 2 ̇c2(t)\n1 −t + 2c2(t)\n(1 −t)2 = 0\n⇒\n ̇c2(t) = −c2(t)\n1 −t = −1\n2 ,\n(38)\nwhich, together with the equations of motion (18) and ρ2 = 0 gives\n−c2(t)\n1 −t = dc2(t)\ndt\n= 3c3(t) −2c2(t)\n1 −t\n⇒\nec3(t) = c3(t)\n1 −t = 1\n3\nc2(t)\n1 −t = 1\n6 .\n(39)\nTherefore the point where the trajectory for α = αa is tangent to the contradiction surface\nbelongs to the critical surface Σcrit. From equation (37) it is clear that since ̇c2 = −1/2,\nthe function tq(α) has infinite slope in (ta, αa), as in figure 3.\n(ii) Next we show that the numerators of the fractions appearing in ̇αd(t) and ̇αs(t) are strictly\npositive if t < tq(α), i.e. in before a contradiction is found. Using the definition (19) we\ncan write:\nF(b; t, α)\n=\nk\nX\nj=1\nρj(t)bj = b\n \n ⟨p1⟩+\nk\nX\nj=2\nbj−2(b −1) ⟨pj⟩\n \n ,\n(40)\nF ′(b; t, α)\n=\nk\nX\nj=1\njρj(t)bj−1 = ⟨p1⟩+\nk\nX\nj=2\nbj−2 [1 −j(1 −b)] ⟨pj⟩.\nThe coefficients in front of ⟨pj⟩≥0 in the sums above are always smaller than 1,\nindependently of j, so that\nF(b; t, α)\n≤\nb\n \n ⟨p1⟩+\nk\nX\nj=2\n⟨pj⟩\n \n ≤b ,\n(41)\nF ′(b; t, α)\n≤\n⟨p1⟩+\nk\nX\nj=2\n⟨pj⟩≤1 .\n(42)\nThe functions F(b; t, α) and F ′(b; t, α) are to be computed in b = bs(t, α) or b = bd(t, α)\nin equations (35) and (36). Both bs and bd are strictly smaller than 1 for all (t, α), as one\ncan directly show from their definitions because V ′(b →1) →∞. Then the coefficients in\nthe sums in (40) are strictly smaller than 1, and the only solution to F = b or F ′ = 1 is\n⟨pj⟩= δ1j, which happens only on the contradiction line.\n2 Recall that we are here talking about average trajectories."},{"page":13,"text":"(iii) The denominators in equations (35), (36) are surely positive at t = 0, as G(b; 0, α) = αbk\nindependently of the heuristic. If they remain positive at all times, then ̇αd(t), ̇αs(t) ≤0 at\nall times, or equivalently dtd\ndα , dts\ndα ≤0 at all α, so that td, ts always increase on decreasing α.\nThe other possibility is that the denominator in (35) crosses zero and become negative,\nleading to a maximum in td(α), which will then decrease on decreasing α. Possibly the\ndenominators can vanish again, giving rise to a sequence of maxima and minima, see right\npanel of figure 3.\nWhat is important is that the numerator is always strictly positive, and as a consequence\ntd(α) or ts(α) are single-valued functions of α. In fact, for td(α) or ts(α) to be multiple-\nvalued functions of α, at some point their slope must become infinite, which is excluded by\nthe analysis above.\n(iv) The statement above, that td(α) and ts(α) are single valued functions of α, implies that\nif a trajectory enters the clustered or unsat phase, it cannot exit from it. This is enough\nto show that αa ≤αd; in fact, the trajectory for α = αa cannot start inside the clustered\nphase, as it would not be able to escape and reach the origin, which is required to find a\nsolution.\n(v) In general the function ec2(t) increases until it reaches a maximum and then decreases\nto 0.\nFor α = αa the value at the maximum is ec2 = 1/2.\nFor α > αa, the value\nat the maximum is ec2 > 1/2, therefore the contradiction ec2 = 1/2 is reached before\nthe maximum, when ec2 is still increasing.\nThen\nd\ndtec2 > 0 at the contradiction point.\nPerforming a simple computation similar to equations (38), (39), one can show that the\ntrajectories for α > αa meet the contradiction surface at ec3 > 1/6.\nNotice then that,\nas it is evident in figure 1, the trajectories corresponding to α > αa must enter first the\nclustered and then the unsat phases in order to reach the contradiction surface, therefore\nfor α < αa one has td(α) < ts(α) < tq(α). On the contrary the trajectories corresponding\nto α < αa must stay away from the clustering and sat/unsat surfaces, otherwise they could\nnot exit and should meet the contradiction surface: therefore for any α < αa, td(α) and\nts(α) do not exist. For α →α+\na , as the surfaces Σd, Σs, Σq are tangent in Σcrit, one has\ntd(αa) = ts(αa) = tq(αa) = ta and the three curves have infinite slope as all the numerators\nin equations (35), (36), (37) vanish on the contradiction surface. This is indeed what is\nobserved in figure 3 for the UC and GUC heuristics, and this argument confirms that this\nis the generic behavior for all the heuristics in the class considered here.\nThis structure is particularly evident for UC, where\nG(UC)(b; t, α) = α[1 −(1 −b)(1 −t)]k −αtk−1[kb(1 −t) + t] .\n(43)\nFrom (43) it is straightforward to check that ∂αG(b; t, α) > 0, ∂αG′(b; t, α) > 0, if b > 0. Then,\nas F(b; t, α) = 2bc2(t)\n1−t\nfor UC, both ̇αd(t) and ̇αs(t) are proportional to 2c2(t)\n1−t −1. This means\nthat αs,αd are decreasing functions of t below the contradiction line.\nThe conclusion is that for a generic Poissonian heuristic, the three lines cross at a critical\npoint (ta, αa) which depends on the heuristic. Above αa the heuristic will cross all the lines\nand find a contradiction. From the properties of the dynamical line, we have that generically\nαa ≤αd, that is no Poissonian search heuristic can find a solution in polynomial time above\nαd, as stated at the beginning of this section. The natural question is then if there exists an\nheuristic that saturates the bound, i.e. such that αa = αd. From the discussion above it is clear\nthat this is possible only if ̇αd(t) ≡0, i.e. the dynamical line in the (t, α) plane is a straight\nvertical line, which is possible only if the numerator in (35) is identically vanishing."},{"page":14,"text":"5.3. Optimality of GUC\nIt is quite easy to see that GUC is the heuristic that locally optimizes the numerator in (35).\nIndeed, from the definition F ′(b; t, α) = Pk\nj=1 jbj−1ρj and the bound F ′(1, t) ≤1, it is clear\nthat F ′(b; t, α) is maximized by maximizing ρj for the smallest possible j, i.e. by picking clauses\nfrom the shortest possible ones, that is GUC. Unfortunately a general proof of the optimality\nof GUC for finite k seems difficult, because one should prove that GUC optimizes globally the\nclustering line, and also control the denominator in (35). In this section we will show that for\nk →∞, GUC is optimal in the sense that ̇αd ≡0 and αd = αa at leading order in k.\nFrom the definition γ(t) = −Pk\nj=2 cj(t) and integrating over time the bound (27), we have\nfor GUC:\nα −\nZ t\n0\ndt′\nj∗(t′) −1 ≤−γ(t) ≤α −\nZ t\n0\ndt′\nj∗(t′) .\n(44)\nor, equivalently,\nα −\nX\nj\nt∗(j) −t∗(j + 1)\nj −1\n≤−γ(t) ≤α −\nX\nj\nt∗(j) −t∗(j + 1)\nj\n.\n(45)\nwhere the sums are limited to the values of j that are reached during the search. In the large k\nlimit, provided the hypothesis\nt∗(j) −t∗(j + 1) = 1\nk + o(1/k)\n(46)\nholds for most j, we obtain\n−γ(t) ≃α −1\nk\nk\nX\nj∗(t)\n1\nj .\n(47)\nThe hypothesis (46) is well supported by numerical data, as shown in figure 4. As the sum of the\ninverse of the first k integers is equivalent to log k (harmonic number) we see that the minimal\nvalue of j∗over t is much larger than 2 if α is much smaller than log k/k. Therefore\nαa ≥log k\nk\n.\n(48)\nThe r.h.s. of the above inequality coincides with the asymptotic scaling of the clustering critical\nratio (section 3.2). Since the results of the previous section require that αa ≤αd, we obtain\nthat α(GUC)\na\n= αd ≃log k/k at the leading order in k →∞. As a comparison, it is easy to see\nthat for UC the threshold for large k is α(UC)\na\n≃e/k, which is therefore much lower than the\nthreshold for GUC.\nThese arguments are supported by numerical simulations that we performed up to k = 216,\nin which the equations of motion (24) are integrated as finite differences equations for all values\nof j (see figure 4). The numerical investigation confirms that kα(GUC)\na\nis very well fitted by\nlog k + 2.15 for k in the range 28 ÷ 216. Moreover, a finite size scaling analysis (with respect to\nk) of the data shown in figure 4 shows that\nk[t∗(j) −t∗(j + 1)] = 1 + kν × f(j/k)\n(49)\nwhere f(x) is a function independent on k which behaves as x−μ for x close to 0. From the\nnumerical data, it appears that ν = μ = 0.5, which confirms that the first correction to the\nleading term log k/k is of order 1/k."},{"page":15,"text":"0\n0.2\n0.4\n0.6\n0.8\n1\nj k\n0.7\n0.8\n0.9\n1\n1.1\n1.2\n1.3\nk Dt*\n5\n6\n7\n8\n9\n10\n11\n12\nlogHkL\n7\n8\n9\n10\n11\n12\n13\n14\nk Αa\n0\n0.2\n0.4\n0.6\n0.8\n1\nj k\n-7.5\n-5\n-2.5\n0\n2.5\n5\n7.5\n10\n !!!k Hk Dt - 1L\n-5\n-4\n-3\n-2\n-1\nlogHxL\n0.5\n1\n1.5\n2\n2.5\n3\nlogI !!!k Hk Dt - 1LM\nFigure 4. Finite size scaling results for GUC at large k. Top Left Each curve shows the values\nof k[t∗(j) −t∗(j + 1)] as a function of j/k for k = 28, 29, . . . , 216 (from the farthest to the closest\ncurve to 1), and was obtained by integrating the equations of motion (24) by finite differences.\nFor each k, the value of α used is αGUC\na\n(k), determined as the value of α for which the maximum\nreached by 2c2(t)/(1 −t) is 1. Top Right Data points of αGUC\na\n(k) versus log k/k + 2.15/k (full\nred line). Bottom left The same data as above, plotted as {k × [t∗(j) −t∗(j + 1)]} × k1/2. The\ncurves “collapse”, showing f(x) and confirming the value of ν = 1/2. Bottom right By plotting\nthe same curves on logarithmic scale it is easily seen that for x close to 0 f(x) ≃x−μ with\nμ = 1/2, corresponding to the slope of the full red line.\n6. Conclusions\nOne of the main results of this paper, that is, that linear-time search heuristic are not able to\nsolve instances in the clustered phase of UE-CSP problems should be interpreted with care. In\nXORSAT-like models the clustering transition coincide with the emergence of strong correlations\nbetween variables in solutions, while the two phenomena generally define two distinct critical\nratios for other random decision problems [15, 16]. From an intuitive point of view it is expected\nthat the performances of search heuristics are affected by correlations between variables rather\nthan the clustering of solutions. Indeed, as the search algorithms investigated here do not allow\nfor backtracking or corrections of wrongly assigned variables, very strong correlations between\nO(N) variables (recall that the backbone includes O(N) variables in the clustered phase) are\nlikely to result in e−O(N) probabilities of success for the algorithm.\nExtending the present work to the random Satisfiability (k-SAT) problem would be interesting\nfrom this point of view, because even if the clustering and freezing transition coincide at leading"},{"page":16,"text":"order for k →∞[3], their finite k values are different in this case. Moreover, in some similar\nproblems (k-COL [17] and 1-in-k-SAT [18]) it has been proven that search algorithms similar to\nthe ones investigated here are efficient beyond the point where the replica-symmetry-breaking\nsolution is stable.\nTherefore these algorithms might beat the clustering threshold in these\nproblems. Note however that in these cases the transition is continuous, so that the structure\nof the clusters is expected to be very different from the one of XORSAT.\nIn addition, while the Generalized Unit Clause heuristic is here shown to be optimal for the\nk-XORSAT problem and to saturate the clustering ratio when k →∞, it is certainly not the\ncase of the k-SAT problem. Determining a provably optimal search heuristic for this problem\nremains an open problem.\n[1] M ́ezard M, Parisi G and Virasoro M A 1987 Spin glass theory and beyond (Singapore: World Scientific)\n[2] Biroli G, Monasson R and Weigt M 2000 Eur. Phys. J. B 14 551\n[3] Krzakala F, Montanari A, Ricci-Tersenghi F, Semerjian G and Zdeborov ́a L 2007 Proc. Natl. Acad. Sci. USA\n104 10318\n[4] Montanari A and Semerjian G 2006 J. Stat. Phys. 124 103\n[5] Monasson R 2007 Introduction to Phase Transitions in Random Optimization Problems, Lecture Notes of\nthe Les Houches Summer School on Complex Systems, Elsevier\n[6] Krzakala F and Kurchan J 2007 Phys. Rev. E 76 021122\n[7] Connamacher H 2004 A Random Constraint Satisfaction Problem That Seems Hard for DPLL, Proceedings\nof the Seventh International Conference on Theory and Applications of Satisfiability Testing\n[8] Chao M T and Franco J 1990 Information Science 51 289\nChao M T and Franco J 1986 SIAM Journal on Computing 15 1106\n[9] Achlioptas D 2001 Theor. Comp. Sci. 265 159\n[10] Achlioptas D, Beame P and Molloy M 2004 J. Comput. Syst. Sci. 68 238\n[11] Dubois O and Mandler J 2002 The 3-XORSAT Threshold, Proceedings of the 43rd Symposium on Foundations\nof Computer Science\n[12] M ́ezard M, Ricci-Tersenghi F and Zecchina R, 2003 J. Stat. Phys. 111 505\n[13] Cocco S, Dubois O, Mandler J and Monasson R 2003 Phys. Rev. Lett. 90 047205\n[14] Weigt M 2002 Eur. Phys. J. B 28 369\n[15] Semerjian G 2007 On the freezing of variables in random constraint satisfaction problems Preprint\narXiv:cond-mat/07052147 (J.Stat.Phys. in press)\n[16] Krzakala F and Zdeborov ́a L 2007 Phys. Rev. E 76 031131\n[17] Achlioptas D and Moore C 2003 J. Comput. Syst. Sci. 67 441\n[18] Raymond J, Sportiello A and Zdeborov ́a L 2007 Phys. Rev. E 76 011101"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"is a collection of M = αN clauses, each involving k variables (out of the N available ones). A","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"corresponds to d = 2 and is solvable in polynomial time with standard linear algebra techniques.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"For d = 3 the problem is still in P, while for d ≥4 it has been shown that (3, d)-UE-CSP is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"propagation implies that if C1 ̸= 0, then pj = δj,1. We consider also the possibility that the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"j=1 pj ≤1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"random, irrespectively of the clauses they appear in (if any), so that pj = 0 unless there are","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"unit clauses; GUC corresponds to pj = δj,j∗where j∗is the length of the shortest clause in the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"incompatible unit-clauses have been generated e.g. x = 0 and x = 1. In the latter case the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"j }j=2,···,k clauses of length j, assuming that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"Cj(T + 1) −Cj(T) = −","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"j′=2 j′Cj′(T)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"Nl(T + 1) −Nl(T) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"j=2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"j′=2 j′Cj′(T)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"l′=0 l′Nl′(T)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"Cj, Nlinto differential equations for their intensive counterparts cj = Cj/N, nl= Nl/N as","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"functions of the reduced number of steps, τ = T/N. The outcome is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"(j = 2, . . . , k) ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"j=2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"where N(τ) = Pk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"j=2 jcj(τ) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"cj(0) = C0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"nl(0) = e−λ0 (λ0)l","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"llnl(0) = λ0 = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"It is easy to check that equations (3) are solved by cj(τ) = cj(0) b(τ)j provided N","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"dτ = −1.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"G(b) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"j=2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"N(τ) = b(τ)G′(b(τ)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"In addition, we define γ(τ) = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"j cj(τ) = G(b(τ)).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"dτ = N","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"dτ = −1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"j=2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"Substituting nl(τ) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"dτ = −","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"b=b(τ)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"with the initial condition imposed by λ(0) = λ0 = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"j jcj(0) = G′(1).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"db = −G′(b) so that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"db = dλ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"db = G′′(b)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"λ(b) = G′(b) ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"The leaf removal algorithm starts from b = 1, then b decreases","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"according to (7) and the algorithm stops at the largest value of b such that n1 = 0, i.e.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"j=2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"lnl= bG′(b) −","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"= λ(b)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"n1 = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"1 −b = e−λ(b) = e−G′(b) .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"This equation always has the solution b = 0, that gives cj = 0 for all j when the algorithm","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"j = αδj,k, the critical ratio at which clustering","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"the condition n1 = 0, the satisfiability condition is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"j}j=2,···,k, or equivalently by the generating function (6). In the following","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"V (b) = −G(b) + b + (1 −b) log(1 −b) .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"The condition n1 = 0 (11), is equivalent to V ′(b) = 0. Thus, if V (b) has a single minimum in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"b = 0, the solution space is not clustered, while if there is another minimum at b ̸= 0, there","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"Σs ≡{cj : V (b) = 0 and V ′(b) = 0 admit a solution b > 0} .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"Σd ≡{cj : V ′(b) = 0 and V ′′(b) = 0 admit a solution b > 0} .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"have dimension k −2 and are surfaces in the space {cj}j=2,···,k of dimension k −1. Note that","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"2c2 = 1, or equivalently V ′′(0) = 0, that defines the contradiction surface Σq:","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"Σq ≡{cj : V ′′(0) = 0} .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"on the region cj ∈[0, 1] for j = 3, . . . , k and c2 ∈[0, 1/2]. Indeed, if one of the cj > 1, the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq64","equation_number":null,"raw_text":"Examples of the phase diagram are in figure 1 for k = 3 and k = 4. There are some special","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq65","equation_number":null,"raw_text":"(i) Recall that Σq is defined by V ′′(0) = 0 and note that V ′(0) = 0 for all b, cj. Thus, on","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq66","equation_number":null,"raw_text":"Σq, the point b = 0 is a solution of both equations (14) and (15). The surfaces Σs, Σd are","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq67","equation_number":null,"raw_text":"values of {cj} the solution with b > 0 merges with the solution b = 0. This happen when","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq68","equation_number":null,"raw_text":"V ′′′(0) = 0, as this is the limiting case in which a saddle at b = bd > 0 and a secondary","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq69","equation_number":null,"raw_text":"minimum at b = bs > 0 can merge for bd, bs →0. The condition V ′′′(0) = 0 is equivalent to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq70","equation_number":null,"raw_text":"c3 = 1/6, and this defines the k −3-dimensional surface","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq71","equation_number":null,"raw_text":"Σcrit ≡{cj : c2 = 1/2, c3 = 1/6} ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq72","equation_number":null,"raw_text":"show that one must consider a displacement c3 = 1/6 + ε and show that (15), (14) admit a","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq73","equation_number":null,"raw_text":"Figure 1. (Left) Schematic phase diagram of k=4-UE-CSP. The full (black) curve is the surface","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq74","equation_number":null,"raw_text":"defined by c2 = 1/2 and c3 = 1/6 and represented as a dashed (blue) line. (Right, top and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq75","equation_number":null,"raw_text":"bottom) The sections of Σd (full, black) and of Σs (dot-dashed, red), at fixed c2 (= 0, 0.1, 0.2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq76","equation_number":null,"raw_text":"0.3, 0.4, 0.5 from top to bottom) as a function of c3 on the top panel, and at fixed c4 (= 0, 0.1,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq77","equation_number":null,"raw_text":"corresponding to c4 = 0 also represent the phase diagram of 3-UE-CSP.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq78","equation_number":null,"raw_text":"for k = 3 but happens for k = 4 for large c4, see figure 1. In this case the transition is called","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq79","equation_number":null,"raw_text":"The generic phase diagram for all k has the shape of the one for k = 4 which we report in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq80","equation_number":null,"raw_text":"variables conditioned to the numbers Cj(T) of clauses of length j (= 2, ..., k) [8, 9].","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq81","equation_number":null,"raw_text":"limit, allows us to study the evolution of the average clauses densities cj(t) = Cj(T)/N on the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq82","equation_number":null,"raw_text":"time scale t = T/N (fraction of assigned variables), which defines a trajectory in the cj’s space.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq83","equation_number":null,"raw_text":"reduced system of N −T variables are ecj(t) = cj(t)/(1 −t). The trajectory of the ecj(t) moves","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq84","equation_number":null,"raw_text":"Initially we have cj(0) = α δjk, i.e. the evolution starts on the ck axis at ck = α. The","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq85","equation_number":null,"raw_text":"̇cj = (j + 1)cj+1 −jcj","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq86","equation_number":null,"raw_text":"ρj(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq87","equation_number":null,"raw_text":"T=tN","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq88","equation_number":null,"raw_text":"average over ∆T defined in (19). Here pk+1 ≡0. Note that the case j = 1 is special as all","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq89","equation_number":null,"raw_text":"j=2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq90","equation_number":null,"raw_text":"̇cj(t) = 2c2(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq91","equation_number":null,"raw_text":"j=2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq92","equation_number":null,"raw_text":"ρj(t) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq93","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq94","equation_number":null,"raw_text":"ρj →0+ for all j ≥2. For this reason we called the plane ρ1 = 1, i.e. ec2 = 1/2, contradiction","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq95","equation_number":null,"raw_text":"In the UC heuristic variables are chosen at random when there is no unit clause. Hence ρj = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq96","equation_number":null,"raw_text":"for j = 2, · · · , k. The solution to (18) is cj(t) = α","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq97","equation_number":null,"raw_text":"probability that the algorithm finds a solution vanishes: for k = 3, α(UC)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq98","equation_number":null,"raw_text":"the continuum limit cj = 0 for j smaller than a given value; therefore we define","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq99","equation_number":null,"raw_text":"j∗(t) = min{j : cj(t) > 0} ,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq100","equation_number":null,"raw_text":"t∗(j) = min[t : cj−1(t) > 0]","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq101","equation_number":null,"raw_text":"the situation in which Cj = 0 for all j < j∗; for this reason ρj∗is not given exactly by 1/j∗.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq102","equation_number":null,"raw_text":"̇cj(t) = (j + 1)cj+1(t) −jcj(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq103","equation_number":null,"raw_text":"Comparing with (18) above, we observe that in the interval t ∈[t∗(j+1), t∗(j)], where j∗= j,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq104","equation_number":null,"raw_text":"ρj∗= 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq105","equation_number":null,"raw_text":"ρj∗−1 = j∗cj∗(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq106","equation_number":null,"raw_text":"j∗(t) ≤ ̇γ(t) = (ρj∗+ ρj∗−1) ≤","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq107","equation_number":null,"raw_text":"variable ratio α. The formula, in the cj space, starts on the axis ck at ck = α. The evolution","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq108","equation_number":null,"raw_text":"of the formula under the action of the algorithm is represented by a trajectory {cj(t, α)}j=2,···,k","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq109","equation_number":null,"raw_text":"or equivalently by G(b; t, α) = Pk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq110","equation_number":null,"raw_text":"j=2 bjcj(t, α), that depends on α through the initial condition","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq111","equation_number":null,"raw_text":"G(b; 0, α) = αbk. We define a potential V (b; t, α) by replacing in (13) G(b) →G(b; t, α)/(1 −t);","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq112","equation_number":null,"raw_text":"the normalization (1 −t) is due to the fact that the cj = Cj/N are divided by N instead of","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq113","equation_number":null,"raw_text":"starting at ck = α at time 0 crosses the surfaces Σs, Σd, Σq defined in section 3.3, which we call","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq114","equation_number":null,"raw_text":"t = {0, td = 0.02957, 0.07327, ts = 0.11697, 0.20642} during the evolution of a (3, d)-UE-CSP","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq115","equation_number":null,"raw_text":"formula with α = 0.8 under the UC heuristic. In the unclustered region it is a convex function","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq116","equation_number":null,"raw_text":"of b with a global minimum in b = 0. On the clustering line td it first develops a secondary","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq117","equation_number":null,"raw_text":"j=2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq118","equation_number":null,"raw_text":"ρj(t)bj =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq119","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq120","equation_number":null,"raw_text":"Performing the total derivative with respect to t of the first condition (V ′ = 0) in (15) for","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq121","equation_number":null,"raw_text":"(αd, bd) and using the second condition, V ′′ = 0, we have","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq122","equation_number":null,"raw_text":"dt + ̇V ′ = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq123","equation_number":null,"raw_text":"dt = −","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq124","equation_number":null,"raw_text":"k = 3. The three lines meet at the critical point (ta, αa) at which the algorithm is no more","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq125","equation_number":null,"raw_text":"dt = −","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq126","equation_number":null,"raw_text":"α=αd(t),b=bd(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq127","equation_number":null,"raw_text":"= 1 −b","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq128","equation_number":null,"raw_text":"Now using V ′′(b; t, α) = −G′′(b;t,α)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq129","equation_number":null,"raw_text":"1−b and V ′′(bd, t) = 0 we have 1−b","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq130","equation_number":null,"raw_text":"1−tG′′(b; t, α) = 1 for b = bd","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq131","equation_number":null,"raw_text":"dt = −1 −F ′(b; t, α)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq132","equation_number":null,"raw_text":"α=αd(t),b=bd(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq133","equation_number":null,"raw_text":"dt = −b −F(b; t, α)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq134","equation_number":null,"raw_text":"α=αs(t),b=bs(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq135","equation_number":null,"raw_text":"The equation for the contradiction line is easily derived from its definition ec2(t, α) = c2(t,α)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq136","equation_number":null,"raw_text":"dt = −1 + 2 ̇c2(t, α)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq137","equation_number":null,"raw_text":"α=αq(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq138","equation_number":null,"raw_text":"Σq. The latter trajectory is tangent to Σq when ec2(t) = 1/2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq139","equation_number":null,"raw_text":"dtec2(t) = 0; the solution to","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq140","equation_number":null,"raw_text":"Moreover, ec2(t) = 1/2 implies that ρ1 = 1 which then implies ρj = 0 for all j ≥2, as already","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq141","equation_number":null,"raw_text":"dtec2(t) = d","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq142","equation_number":null,"raw_text":"1 −t = 2 ̇c2(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq143","equation_number":null,"raw_text":"(1 −t)2 = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq144","equation_number":null,"raw_text":"̇c2(t) = −c2(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq145","equation_number":null,"raw_text":"1 −t = −1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq146","equation_number":null,"raw_text":"which, together with the equations of motion (18) and ρ2 = 0 gives","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq147","equation_number":null,"raw_text":"1 −t = dc2(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq148","equation_number":null,"raw_text":"= 3c3(t) −2c2(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq149","equation_number":null,"raw_text":"ec3(t) = c3(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq150","equation_number":null,"raw_text":"1 −t = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq151","equation_number":null,"raw_text":"1 −t = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq152","equation_number":null,"raw_text":"Therefore the point where the trajectory for α = αa is tangent to the contradiction surface","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq153","equation_number":null,"raw_text":"belongs to the critical surface Σcrit. From equation (37) it is clear that since ̇c2 = −1/2,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq154","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq155","equation_number":null,"raw_text":"ρj(t)bj = b","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq156","equation_number":null,"raw_text":"j=2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq157","equation_number":null,"raw_text":"j=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq158","equation_number":null,"raw_text":"jρj(t)bj−1 = ⟨p1⟩+","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq159","equation_number":null,"raw_text":"j=2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq160","equation_number":null,"raw_text":"j=2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq161","equation_number":null,"raw_text":"j=2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq162","equation_number":null,"raw_text":"The functions F(b; t, α) and F ′(b; t, α) are to be computed in b = bs(t, α) or b = bd(t, α)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq163","equation_number":null,"raw_text":"the sums in (40) are strictly smaller than 1, and the only solution to F = b or F ′ = 1 is","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq164","equation_number":null,"raw_text":"⟨pj⟩= δ1j, which happens only on the contradiction line.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq165","equation_number":null,"raw_text":"(iii) The denominators in equations (35), (36) are surely positive at t = 0, as G(b; 0, α) = αbk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq166","equation_number":null,"raw_text":"to show that αa ≤αd; in fact, the trajectory for α = αa cannot start inside the clustered","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq167","equation_number":null,"raw_text":"For α = αa the value at the maximum is ec2 = 1/2.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq168","equation_number":null,"raw_text":"at the maximum is ec2 > 1/2, therefore the contradiction ec2 = 1/2 is reached before","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq169","equation_number":null,"raw_text":"td(αa) = ts(αa) = tq(αa) = ta and the three curves have infinite slope as all the numerators","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq170","equation_number":null,"raw_text":"G(UC)(b; t, α) = α[1 −(1 −b)(1 −t)]k −αtk−1[kb(1 −t) + t] .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq171","equation_number":null,"raw_text":"as F(b; t, α) = 2bc2(t)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq172","equation_number":null,"raw_text":"heuristic that saturates the bound, i.e. such that αa = αd. From the discussion above it is clear","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq173","equation_number":null,"raw_text":"Indeed, from the definition F ′(b; t, α) = Pk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq174","equation_number":null,"raw_text":"j=1 jbj−1ρj and the bound F ′(1, t) ≤1, it is clear","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq175","equation_number":null,"raw_text":"k →∞, GUC is optimal in the sense that ̇αd ≡0 and αd = αa at leading order in k.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq176","equation_number":null,"raw_text":"From the definition γ(t) = −Pk","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq177","equation_number":null,"raw_text":"j=2 cj(t) and integrating over time the bound (27), we have","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq178","equation_number":null,"raw_text":"t∗(j) −t∗(j + 1) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq179","equation_number":null,"raw_text":"= αd ≃log k/k at the leading order in k →∞. As a comparison, it is easy to see","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq180","equation_number":null,"raw_text":"These arguments are supported by numerical simulations that we performed up to k = 216,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq181","equation_number":null,"raw_text":"k[t∗(j) −t∗(j + 1)] = 1 + kν × f(j/k)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq182","equation_number":null,"raw_text":"numerical data, it appears that ν = μ = 0.5, which confirms that the first correction to the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq183","equation_number":null,"raw_text":"of k[t∗(j) −t∗(j + 1)] as a function of j/k for k = 28, 29, . . . , 216 (from the farthest to the closest","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq184","equation_number":null,"raw_text":"curves “collapse”, showing f(x) and confirming the value of ν = 1/2. Bottom right By plotting","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq185","equation_number":null,"raw_text":"μ = 1/2, corresponding to the slope of the full red line.","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":43659,"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}}