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{"paper_meta":{"paper_id":"arxiv:0711.0110","title":"0711.0110","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:0711.0110v1 [cs.CC] 1 Nov 2007\nProceedings of the International Workshop on Statistical-Mechanical Informatics\nSeptember 16–19, 2007, Kyoto, Japan\nPhase Transitions and Computational Difficulty in\nRandom Constraint Satisfaction Problems\nFlorent Krz ֒aka la1 and Lenka Zdeborov ́a2\n1 PCT, UMR Gulliver 7083 CNRS-ESPCI, 10 rue Vauquelin, 75231 Paris, France\n2 LPTMS, UMR 8626 CNRS et Univ. Paris-Sud, 91405 Orsay CEDEX, France\nE-mail: fk@espci.fr\nAbstract.\nWe review the understanding of the random constraint satisfaction problems,\nfocusing on the q-coloring of large random graphs, that has been achieved using the cavity\nmethod. We also discuss the properties of the phase diagram in temperature, the connections\nwith the glass transition phenomenology in physics, and the related algorithmic issues.\n1. Introduction\nSpin glass theory has a large and probably initially unexpected impact on some problems far\nfrom condensed matter physics and one example of such spectacular outcome is the application of\nstatistical physics ideas to combinatorial optimization [1] and of the concept of phase transitions\nto the probabilistic analysis of Constraint Satisfaction Problems (CSPs) [2, 3, 4]. Given a set\nof N discrete variables subject to a set of M constraints, a CSP consists in deciding if there\nexists an assignment of the variables satisfying all the constraints. This is a generic setting that\nis currently used to tackle problems as diverse as, among others, error-correcting codes, register\nallocation in compilers or genetic regulatory networks. The class of NP-complete problems [1],\nfor which no algorithm is known that guarantees to decide satisfiability in a time polynomial in\nN, is particularly interesting. Well-studied examples of such problems are the satisfiability of\nboolean formulas (SAT), and the q-coloring problem (q-COL, see figure 1) that we shall discuss\nhere. Given a graph with N vertices and M edges connecting certain pairs of them, and given\nq colors, can we color the vertices so that no two connected vertices have the same color?\nCrucial empirical observations were made when considering the ensemble of random graphs\nwith a given average vertex connectivity c: while below a critical value cs a proper q-coloring\nof the graph exists with a probability going to one in the large size limit, it was found that\nbeyond cs no proper q-coloring exists asymptotically. This sharp threshold (which appears in\nother CSPs such as K-SAT and whose existence is partially proved in [5]) is an example of a\nphase transition arising in random CSPs. It was also observed empirically [6, 7] that deciding\ncolorability becomes on average much harder near to the coloring threshold cs than far away\nfrom it. It is therefore natural to ask ourselves: Can the value of the colorable/uncolorable\n(COL/UNCOL) phase transition be computed? Can the number of all possible colorings be also\ncomputed? Are there other interesting phase transitions? Can these transitions explain the fact\nthat solutions are sometimes very hard to find? Can this knowledge help us in designing new\nalgorithms? These questions, and their answers, are at the roots of the interest of the statistical\nphysics community in optimization problems [3, 4].\n\n2. A Potts anti-ferromagnet on random graphs\nIt is immediate to realize that the q-coloring problem is equivalent to the question of determining\nif the ground-state energy of a Potts anti-ferromagnet on a random graph is zero or not [8].\nConsider indeed a graph G = (V, E) defined by its vertices V = {1, . . . , N} and edges (i, j) ∈E\nwhich connect pairs of vertices i, j ∈V; and the Hamiltonian\nH({s}) =\nX\n(i,j)∈E\nδ(si, sj) .\n(1)\nWith this choice there is no energy contribution for neighbors with different colors,\nbut a positive contribution otherwise.\nThe ground state energy (the energy at zero\ntemperature) is thus zero if and only if the graph is q-colorable.\nThis transforms the\ncoloring problem into a well-defined statistical physics model. Usually, two types of random\ngraphs are considered:\nin the c−regular ensemble all points are connected to exactly\nc neighbors, while in the Erd ̋os-R ́enyi case the connectivity has a Poisson distribution.\nFigure 1. Example: a proper 3-\ncoloring of a small graph.\n3. Cavity method: Warnings, Beliefs and Surveys\nOver the last few years,\na number of studies have\ninvestigated CSPs following the adaptation of the so-called\ncavity method [2] to random graphs [4, 9]. It is a powerful\nheuristic tool —whose exactness is widely accepted but\nhas still to be rigorously demonstrated— equivalent to the\nreplica method of disordered systems [2].\nIts main idea\nlies in the fact that a large random graph is locally tree-\nlike, and that an iterative procedure known in physics as\nthe Bethe-Peirls method can solve exactly any model on\na tree (such models are often qualified as “mean field”\nin physics).\nInterestingly, it was realized [10] that an\nequivalent formalism has been developed independently in\ncomputer science [11], where it is called Belief Propagation\n(BP, which conveniently enough, may also stands for\nBethe-Peirls). Defining ψi→j\nc\n(c = 1, .., q) as the probability that the spin i has color c in absence\nof the spin j (the “belief” that the spin j has on the properties of the spin i), BP reads\nψi→j\nc\n=\n1\nZi→j\n0\nY\nk∈N(i)\\{j}\n \n1 −ψk→i\nc\n \n(2)\nwhere Zi→j\n0\nis a normalization constant and the notation k ∈N(i) \\ {j} means the set of\nneighbors of i except j. From a fixed point of these equations, the complete beliefs in presence\nof all spins can be also computed. They give, for each vertex, the probability of each color from\nwhich other quantities, as for instance the number of solutions, can be computed. A simpler\nformalism, called Warning Propagation (WP), restricts itself to frozen variables (i.e. to variables\nfor which only one color can satisfy the constraints). However, WP does not allow to compute\nthe number of solutions, only their existence, but is definitely simpler to handle.\nIt was soon realized, however, that these methods developed for trees could not be\nused straightforwardly on all random graphs because of a non-trivial phenomenon called\nclustering [9, 12] (for which rigorous results are now available, see [13]).\nIndeed, while for\ngraphs with very low connectivities all solutions are “connected” —in the sense that it is easy\nwith a local dynamics to move from one solution to another— they regroup into a large number\n\nof disconnected clusters for larger connectivities. It can be argued that each of these clusters\ncorresponds to a different fixed point of the BP equations, so that a survey over the whole set of\nthe fixed points should be performed. This can be done in the cavity method by the now famous\nSurvey Propagation (SP) equations [4] which, in the physics language, correspond to the Parisi’s\none-step Replica Symmetry Breaking (RSB) scheme [2]. Within this formalism, the number of\nclusters (which behaves as N = eNΣ, where Σ is a fundamental quantity called the complexity)\nand their sizes (the number of proper colorings inside the cluster) can be determined.\nThis formalism has been applied on the SAT [4] and COL [14, 15] problems in the limit\nof infinitely large graphs. These cutting edge studies were however restricted to SP applied\nto the clusters corresponding to fixed points of WP and not to those of BP. Although this\nalready allowed the correct computation of the COL/UNCOL transition and the development\nof a powerful algorithm [4], it meant that the description of the clustered phase was only partial\nand this resulted in a number of problems and inconsistencies that stayed unanswered until very\nrecently. These issues have been today clarified [16, 17, 18, 19, 20, 21] and we shall now discuss\nthis new understanding.\n4. The phase diagram of the coloring problem on a random graph\nConsider that we have q ≥4 colors (the q = 3 case being a bit particular [16, 18, 20], as\nwe shall see) and a large random graph whose connectivity c we shall increase.\nDifferent\nphases are encountered that we will now describe (and enumerate) in order of appearance (the\ncorresponding phase diagram is depicted in figure 2).\n(i) A unique cluster exists: For low enough connectivities, all the proper colorings are\nfound in a single cluster, where it is easy to “move” from one solution to another. Only\none possible —and trivial— fixed point of the BP equations exists at this stage (as can be\nproved rigorously in some cases [23]). The entropy can be computed and reads in the large\ngraph size N limit\ns = log Nsol\nN\n= log q + c\n2 log\n \n1 −1\nq\n \n.\n(3)\n(ii) Some (irrelevant) clusters appear: As the connectivity is slightly increased, the phase\nspace of solutions decomposes into an large (exponential) number of different clusters. It\nis tempting to identify that as the clustering transition, but it happens that all (but one)\nof these clusters contain relatively very few solutions —as compare to whole set— and that\nalmost all proper colorings still belong to one single giant cluster. Clearly, this is not a\nproper clustering phenomenon and in fact, for all practical purpose, there is still only one\nsingle cluster. equation (3) still gives the correct entropy at this stage.\n(iii) The clustered phase: For larger connectivities, the large single cluster also decomposes\ninto an exponential number of smaller ones: this now defines the genuine clustering threshold\ncd1. Beyond this threshold, a local algorithm that tries to move in the space of solutions\nwill remain prisoner of a cluster of solutions [24]. Interestingly, it can be shown that the\ntotal number of solutions is still given by equation (3) in this phase. This is because, as\nis well known in the replica method, the free energy has no singularity at the dynamical\ntransition (which is therefore not a true transition in the sense of Ehrenfest, but rather a\ndynamical or geometrical transition in the space of solutions).\n(iv) The condensed phase: As the connectivity is further increased, a new sharp phase\ntransition arises at the condensation threshold cc where most of the solutions are found in\n1 It is important to point out that the location of the clustering transitions was therefore not computed correctly\nwhen the dependence on the size of clusters was not taken into account. Also, different results were obtained\npreviously depending on whether or not unfrozen fixed points were explicitly considered.\n\nCluster without frozen variable\nCluster with frozen variables\nCONDENSATION\nRIGIDITY\nUNCOL\nconnectivity\nUncolorable phase\nColorable phase\nCLUSTERING\nc\nc\nc\nc\nc\ns\nr\nd\n(i)\n(ii)\n(iii)\n(iv)\n(v)\n(vi)\nFigure 2. Sketch of the space of solutions —colored points in this representation— in the\nq-coloring problem on random graphs when the connectivity c is increased. (i) At low c, all\nsolutions belong to a single cluster. (ii) For larger c, other clusters of solutions appear but a\ngiant cluster still contains almost all solutions. (iii) At the clustering transition cd, it splits\ninto an exponentially large number of clusters. (iv) At the condensation transition cc, most\ncolorings are found in the few largest of them. (v) The rigidity transition cr (cr < cc and cr > cc\nare both possible depending on q) arises when typical solutions belong to clusters with frozen\nvariables (that are allowed only one color in the cluster). (vi) No proper coloring exists beyond\nthe COL/UNCOL threshold cs.\na finite number of the largest clusters. From this point, equation (3) is not valid anymore\nand becomes just an upper bound. The entropy is non-analytic at cc therefore this is a\ngenuine static phase transition.\n(v) The rigid phase: As mentioned in section 3, two different types of clusters exist: In the\nfirst type, that we shall call the unfrozen ones, all spins can take at least two different colors.\nIn the second type, however, a finite fraction of spins is allowed only one color within the\ncluster and are thus “frozen” into this color. These frozen clusters actually correspond to\nnon-trivial fixed points of BP and WP, while the first kind are non-trivial fixed points of BP\nonly. It follows that a transition exists, that we call rigidity, when frozen variables appear\ninside the dominant clusters (those that contains most colorings). If one takes a proper\ncoloring at random beyond cr, it will belong to a cluster where a finite fraction of variables\nis frozen into the same color. Depending on the value of q, this transition may arise before\nor after the condensation transition (see table 1).\n(vi) The UNCOL phase: Eventually, the connectivity cs is reached beyond which no more\nsolutions exist. The ground state energy (sketched in figure 3) is zero for c < cs and then\ngrows continuously for c > cs. The values cs computed within the cavity formalism are in\nperfect agreement with the rigorous bounds [25] derived using probabilistic methods and\nare widely believed to be exact (although they remains to be rigorously proven, but see [26]\nfor a proof that they are at least rigorous upper bounds).\nWe report the values of the threshold connectivities corresponding to all these transitions in\ntable 1 for the regular and the Poissonian (i.e. Erd ̋os-R ́enyi) random graphs ensembles. Notice\nthat the 3-coloring is peculiar because cd = cc so that the clustered phase is always condensed in\nthis case. In view of this rich phase diagram, it is important to get an intuition on the meaning\nand the properties of these different phases and, in this respect, it is interesting before entering\nthe algorithmic implications to discuss the analogies with the glass transition.\n\nq\ncd\ncr\ncc\ncs\n3\n5+\n-\n6\n6\n4\n9\n-\n10\n10\n5\n14\n14\n14\n15\n6\n18\n19\n19\n20\n7\n23\n-\n25\n25\n8\n29\n30\n31\n31\n9\n34\n36\n37\n37\n10\n39\n42\n43\n44\nq\ncd\ncr\ncc\ncs\n3\n4\n4.66(1)\n4\n4.687(2)\n4\n8.353(3)\n8.83(2)\n8.46(1)\n8.901(2)\n5\n12.837(3)\n13.55(2)\n13.23(1)\n13.669(2)\n6\n17.645(5)\n18.68(2)\n18.44(1)\n18.880(2)\n7\n22.705(5)\n24.16(2)\n24.01(1)\n24.455(5)\n8\n27.95(5)\n29.93(3)\n29.90(1)\n30.335(5)\n9\n33.45(5)\n35.658\n36.08(5)\n36.490(5)\n10\n39.0(1)\n41.508\n42.50(5)\n42.93(1)\nTable\n1.\nThreshold\nconnectivities\ncd\n(dynamical/clustering)\n[20,\n17,\n18,\n27],\ncr\n(rigidity/freezing) [20, 21], cc (condensation/Kauzmann) [20, 18] and cs (COL/UNCOL) [14, 15]\nfor regular (left) and Erd ̋os-R ́enyi (right) random graphs. In the large q-limit, one finds in both\ncases that [20, 18]: cr = q[log q + log log q + 1 + o(1)], cc = 2q log q −log q −2 log 2 + o(1) and\n[15]: cs = 2q log q −log q −1 + o(1).\n5. A detour into the ideal glass transition phenomenology\nTo those familiar with the replica theory and the mean field theory of glasses, the phenomenology\ndepicted in the former section should look familiar: these successive transitions are indeed very\nwell known in the picture of the ideal glass transition [28].\nThis striking analogy is in fact\nquite natural since, despite the fact that there is no disorder in the interactions in Hamiltonian\n(1), the frustration due to the loops in the random graph makes the model behaving like a\ndisordered “anti-ferromagnetic” Potts spin glass [29] and such models are known to display the\nglassy phenomenology [8, 28].\nThe phase diagram obtained on Poissonian random graphs with average connectivity c for\nq ≥4 is sketched in figure 3 (the q = 3 model is slightly different as, again, Td = Tc). At\nhigh temperature the system behaves as a liquid (or a paramagnet in the language of magnetic\nsystems). Below a temperature Td a first transition —called “dynamical”— happens and the\nsystem falls out of equilibrium.\nFor T < Td it is not possible for a physical dynamics to\nequilibrate the system and the ergodicity is broken: this is due to the appearance of exponentially\nmany different states.\nHowever, the would-be equilibrium properties of the problem remain\nsimilar (and in particular, the free-energy has no singularity at this temperature).\nOnly at\ntemperature Tc the free energy is non analytic and a true “static” glass transition happens,\ncalled the Kauzmann transition [28]. In this phase only a finite number of states does matter\nat a given connectivity. Finally, for larger connectivities, a third phenomenon is observed as\nthe temperature is further lowered, called the Gardner transition [30, 31]. It is a transition\ntowards a more complicated phase, similar to the one found in the celebrated solution of the\nSherrington-Kirkpatrick model [2]. The fact that the Gardner transition arises for connectivities\nlarger the COL/UNCOL one is very important in this respect: it shows that the study of this\nphase, that requires a more involved cavity formalism (and probably further RSB), does not\nseem to be needed in the colorable phase2. We also now recognize that the “clustering” and the\n“condensation” transitions in the coloring problem are just the zero temperature relics of the\ndynamic and Kauzmann transitions at finite temperatures.\nA similar connection with the physics of glassy system can also be obtained directly at zero\n2 The expert reader might find this puzzling, as many papers stated that the simple “one replica symmetry\nbreaking” [4, 9] solution was unstable towards a more complex solution in some region of the COL/SAT phase\n[32, 15]. However, these results were obtained neglecting the role of the sizes of the clusters: while in some cases\nmost clusters are indeed unstable, our studies [29] indicate that the relevant ones seem always stable in the COL\nphase (although the cases of 3−COL and 3−SAT might be problematic, see [20, 29]).\n\nd\nC\nCG\ne (c)\nGS\nCc\nCs\nEnergy\nConnectivity\nTemperature\nTd\nc\nT\nG\nT\nGardner phase\nLiquid\n\"Dynamic\"\n\"Static\" glass\nglass\nFigure\n3.\nSketch of the\nphase diagram in the color-\ning problem at finite temper-\nature (from [29]). At Td, the\nsystem falls out of equilib-\nrium (“dynamic” transition).\nAt Tc the system undergoes a\n“static” glass transition.\nFi-\nnally, at TG, a Gardner tran-\nsition appears. eGS represents\nthe ground state energy.\ntemperature via the jamming phenomenology [19] where the density of constraints (in this case\nthe volume of some non-overlapping spheres in a box of fixed volume) are increased and where\na dynamical transition is first met while some authorized configurations exist much beyond\nthis point (see for instance [33] and references therein). We thus see that the coloring problem\non random graphs translates into a very general mean field model of a complex liquid. This\nconvergence of interest between different disciplines is quite interesting in itself and allows to\ndiscuss a number of matters, as we shall now see.\n6. Onset of hardness for local search algorithms\nThe properties of the phase diagram we just discussed are based on analytical computations\nthrough the cavity method. We would like to discuss now what are the implications of these\ndifferent phases on the performance of simple local search algorithms that try to find a solution.\nThis is, however, a much harder subject to handle analytically and we shall thus leave the field of\nanalytical computations to enter the one of phenomenology. Still, the behavior of an algorithm\ntrying to find a solution is reminiscent of the behavior of the physical dynamics in glassy systems,\nand we can at least exploit this analogy in order to get an intuition for the problem that we can\nlater confirm with numerical simulations.\nIt is first tempting to identify the point cd, where a physical Monte-Carlo dynamics gets\ntrapped into a cluster, with the onset of computational hardness3. However, a second moment\nthough indicates that this should not be the case: In the glass transition phenomenology, it\nis well known that, although the system falls out of equilibrium beyond Td, its energy can be\nfurther lowered by lowering the temperature, or just by waiting a bit longer [19]. In short: the\nfact that dynamics is prisoner of a given region of the set of possible configurations does not\nmean that no solution can be found in this region. Although cd is indeed a sharp transition for\nthe Monte-Carlo sampling, there is no reason, a priori, to experience difficulties if one just want\nto find one solution beyond this point. This is particularly transparent in the analysis and the\nalgorithm introduced in [19] (and directly inspired from the analogy with jamming [33]):\n(i) Start with a graph of connectivity c and a proper coloring.\n(ii) Increase the density of constraint by adding a link in the graph.\n(iii) Use a simple algorithm in order to solve the contradiction introduced by the link. When it\nis done, go back to step (i).\nBy applying this strategy, starting from scratch (i.e. from a graph with N vertices and no\nlink), the set of all proper colorings undergoes the successive transitions described in figure 2 as\n3 When the clustering phenomena was discovered in CSPs, it was indeed initially conjectured to be responsible\nfor the onset of hardness for local search strategies [12, 4] and the clustered phase was named the “hard phase”.\nHowever, some local algorithms were found to easily beat the threshold (see [34] for SAT and [20] for COL).\n\nZero energy states\nPositive energy states\nFigure\n4.\nArtist’s view of the energy\nlandscape for low connectivities c > cd:\na\nregion dominated by canyons that reach the\nground-states.\nZero energy states\nPositive energy states\nFigure\n5.\nArtist’s view of the energy\nlandscape for large connectivities c > cd: a\nregion dominated by high mountains and deep\nvalleys.\nconnectivity increases. When the dynamical transition is reached, one is trapped inside a cluster\nof solutions, but this is not really a problem as one is still free to move inside the cluster. As\nmore links are added, the cluster size is decreasing continuously but while it still exists, the local\nalgorithm should be in principle able to find solutions nearby. Only for larger connectivities,\nwhen the cluster gets frozen, it disappears and consequently the algorithm stops. It was shown\nin Ref. [19] through numerical simulations that this strategy, using the Walk-COL [20] algorithm\nfor step (iii), is indeed efficient, and linear in N, much beyond the dynamical transition.\nThe reason why this recursive strategy becomes inefficient when the cluster in which the\ndynamics is trapped freezes is the following: if a link is put between two vertices frozen in the\nsame color, it is impossible to satisfy the constraints while remaining in the cluster. As opposed\nto the unfrozen clusters, the frozen clusters thus have a finite probability to disappear when a\nnew link is added. A cavity-like analysis [21], confirmed by numerical data [19], actually shows\nthat the number of changes that the algorithm must perform, in order to solve the contradictions\nimposed by the addition of new links, increases with the connectivity and diverges when the\nfrozen variables appear. The source of difficulties is therefore not the clustering phenomenon in\nitself, but rather the appearance of frozen variables. This makes the analysis and the prediction\nof a EASY/HARD threshold much harder since (as one can see on figure 2) clusters of different\nsizes freeze at different connectivities, although a connectivity c∗≥cr exists where all clusters\nare frozen, thus putting a strict bound to the efficiency of this procedure.\nInterestingly, even non-incremental algorithms may also pass the cd threshold [34, 20], and\nthat might come as a surprise for those having in mind the “rugged” many-valley energy\nlandscape picture of spin glasses.\nThis apparent paradox can be clarified by the following\nconsiderations: It is possible that at lower connectivity c > cd the energy landscape is dominated\nby deep canyons (figure 4), where it is in principle easy to go down as one has just to jump ahead!\nAt larger connectivities a more rugged region with many deep valleys and high mountains is\nfound (figure 5) in which case, as any mountain-hiker will undoubtly know, it takes some time to\ngo to the deepest valley because many hills have to be climbed first. This difference in behavior\nmight explain the “unreasonable efficiency” of local algorithm [7] and the performance of the\nannealing procedure beyond cd [27].\nTo further illustrate this point, consider the Walk-COL algorithm introduced in [20] (and\nadapted from a similar one in SAT [34]) defined by the following procedure\n(i) Randomly choose a spin that has the same color as at least one of its neighbors.\n\n1e-05\n 1e-04\n 0.001\n 0.01\n 0.1\n 1000\n 10000\n 100000\n 1e+06\n 1e+07\nFraction of unsatisfied variables\nt/N\nc=8.0\nc=8.3\nc=8.4\nc=8.5\nc=8.75\nN=50 000\nN=200 000\n 1000\n 10000\n 100000\n 1e+06\n 8\n 8.5\n 9\n \n \nτ(c)\ncd\ncc\ncr cs\nFigure 6.\nFraction of unsatis-\nfied variables versus the number\nof attempted flips of the Walk-\nCOL algorithm divided by the\nsize of the graph in the q = 4 col-\noring: Walk-COL [20] algorithm\nis able to find some solutions in\nthe clustered phase for low c (in\nthe canyon-like region), but get\ntrapped in the high energy val-\nleys for larger c. Inset: estimated\ntime τ = t/N needed to find a\nsolution versus connectivity.\n(ii) Change randomly its color.\nAccept this change with probability one if the number of\nunsatisfied spins has been lowered, otherwise accept it with probability p (this is a parameter\nthat has to be tuned for better efficiency).\n(iii) If there are unsatisfied vertices, go to step (i) unless the maximum running time is reached.\nThis algorithm can easily find colorings for large sizes in linear time beyond cd [20], but certainly\nnot too close to the UNCOL transition where it gets trapped at higher energies (see figure 6).\nSo far, there are few analytical results about the energy landscape in this problem and it\nis likely that this will be the subject of further studies. It is unfortunately very hard to say\nfor which connectivities the landscape goes from canyons-dominated to mountains-dominated\nas this may not be a sharp transition and more a matter of —certainly algorithm-dependent—\nbasins of attraction. The rigidity transition for typical clusters is certainly a good candidate as\na crossover in behavior (as is the connectivity where all clusters are frozen).\nTo conclude, one sees that although the algorithmic issues are indeed more difficult to handle\nthan the phase diagram, at least two important points can already be made: First, the dynamical\ntransition does not correspond to the onset of hardness, and second, the source of difficulty seems\nmore to be related with the appearance of frozen variables.\n7. Message Passing and Decimation\nThe class of local search algorithms is only one part of the story.\nA different class, where\nmessages are exchanged through the nodes of the graph, was proven to be very efficient in\ncomputer science. BP and WP are examples of such procedures and it is thus interesting to\nuse the information given by the cavity analysis to discuss their performance in estimating the\nmarginals —or other informations— in the problem. A major outcome of the last years has also\nbeen the application of SP as a message passing (MP) [4].\nFrom the information given by the fixed point of the MP, an algorithm can be defined in\nthe following way [4]: (i) run the MP to obtain some information, and (ii) fix (decimate) some\nvariables according to this information. This sequence is iterated until a solution is found, or\nuntil a contradiction is met. The two parts are quite independent and each of them can be\nchanged separately (for instance a self consistent re-enforcement has been tried instead of the\ndecimation with very good results [35]).\nAccording to the cavity formalism, the iterative fixed point of BP correctly estimates the\nmarginals until the condensation at cc [18] (which, conveniently, is very close to cs). Indeed\n\nthe application of BP plus decimation is numerically efficient in finding solutions for both SAT\nand COL [18, 20, 22] and it would be interesting to see if this method allows to find “typical”\nsolutions beyond cd, thus bypassing the problems of Monte-Carlo algorithms. It seems that\nthe strategy is efficient even beyond cc in some cases [20], although the BP recursion does not\nalways converge (and when it does, it does it slowly) on the decimated graph, in which case\nan imprecise and approximate information is obtained. These issues are thus far from being\nproperly understood at the present time, and we hope that new works will be done in this\ndirection.\nA more powerful strategy in the clustered phase is to use SP to compute the probability\nthat a given variable is frozen is a cluster [4]. Fixing the variables which are frozen in most\nclusters seems a good way to decimate the graph. Using this method in random 3-SAT allows to\noutperform any other known algorithm and to find solutions of huge instances rapidly even very\nclose to the threshold [4, 35]. The method has been subsequently adapted for the 3−coloring in\n[14]. Interestingly, SP behaves better than BP on the decimated graph and has no problem of\nconvergence. Other strategies are possible that have not yet been exploited [20].\nThe best way to extract the informations given by the fixed point of these message passing\nprocedures is however not yet known, nor is the limit until where these strategies are efficient.\nHowever, this line of thoughts is certainly the most promising way in the direction of better\nsolving and sampling algorithms.\n8. Conclusions and perspectives\nIn this paper, we considered and reviewed some aspects of the q−coloring of large random graphs.\nWe discussed the properties of the set of solutions and the different sharp phase transitions it\nundergoes when the average connectivity is varied. The problem translates in physics into a\nmean-field model with an ideal glass transition. The cavity method has been efficient in giving\ninsight into the problem, although the important challenge of proving rigorously these results\nremains.\nIt would be interesting in this respect to confirm these predictions by performing\nextensive Monte-Carlo simulations of the phase diagram depicted in figure 3.\nWe also discussed the dynamical behavior of local search algorithms. We saw that although\nthe “dynamical” transition has a direct meaning as the point where local Monte-Carlo algorithms\nget out of equilibrium, it is not directly connected with the onset of hardness in the problem\nwhich is rather due to the fact that clusters “freeze” as the connectivity increases. So far, it has\nnot therefore been possible, despite initial hope, to have well-defined HARD and EASY phases.\nThe precise role of frozen variables and the part played by the rigidity transition, or by the\nconnectivity where all clusters becomes frozen, will undoubtly be the subject of new research,\nboth in numerical and theoretical directions, that will help to clarify these issues. A refined\nknowledge of the energy landscape would also be valuable.\nFinally, we also discuss the major breakthrough in the algorithmic strategy that emerged\nfrom the application of the cavity solution on a single given graph. It is not clear at the present\ntime what it the best way to use this approach and how efficient it can be, nor if it is possible to\ngo arbitrarily close to the satisfiability/colorability threshold for any values of q, and it is likely\nthat these questions will also trigger a lot of work in the future.\nAcknowledgments\nWe thank J. Kurchan, A. Montanari, F. Ricci-Tersenghi and G. Semerjian for the collaborations\nthat led to a substantial part of our results. We also greatly benefit from discussions with M.\nM ́ezard and R. Zecchina. We thank T. J ̈org and A. Hartmann for a critical lecture of the paper.\n\nReferences\n[1] Cook S 1971 Proc. of the 3rd Annual ACM Symp. on Theory of Computing (ACM, New York) pp. 151-\n158. Papadimitrious C H 1994 Computational Complexity (Addison-Wesley, Reading, MA). Garey M R\nand Johnson D S 1979 Computers and Intractability (Freeman, San Francisco). Dubois O, Monasson R,\nSelman B, and Zecchina R 2001 special issue of Theor. Comp. Sci. 265. 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Achlioptas D, Naor A and Peres Y 2005 Nature 435 759.\n[26] Franz S and Leone M 2003 J. Stat. Phys. 111 535.\n[27] van Mourik J and Saad D 2002 Phys. Rev. E 66 056120.\n[28] Gross D J and M ́ezard M 1984 Nuclear Physics B 240 4 431. Kirkpatrick T, Thirumalai D, and Wolynes P\n1989 Phys. Rev. A 40 1045. M ́ezard M and Parisi G 1999 Phys. Rev. Lett. 82 747.\n[29] Krzakala F and Zdeborov ́a L 2007 Potts Glass on Random Graphs Preprint arXiv:0710.3336.\n[30] Gardner E 1985 Nuclear Physics B 257 747.\n[31] Leuzzi L and Parisi G 2001 J. Stat. Phys. 103 679. Crisanti A, Leuzzi L and Parisi G 2002 J. Phys. A:\nMath. Gen. 35 481.\n[32] Montanari A, Parisi G and Ricci-Tersenghi F 2004 J. Phys. A: Math. Gen. 37 2073. Mertens S, M ́ezard M\nand Zecchina R 2003 Random Struct. Algorithms 28 3 340.\n[33] Liu A and Nagel S 1998 Nature 396 21. O’Hern C S, Silbert L E, Liu A and Nagel S 2003 Phys. Rev. E 68\n011306. Wyart M 2005 Annales de Physique 30 3.\n[34] Ardelius J and Aurell E 2006 Phys. Rev. E 74 037702.\n[35] Chavas J, Furtlehner C, M ́ezard M, and Zecchina R 2005 J. Stat. Mech. P11016.","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0711.0110v1 [cs.CC] 1 Nov 2007\nProceedings of the International Workshop on Statistical-Mechanical Informatics\nSeptember 16–19, 2007, Kyoto, Japan\nPhase Transitions and Computational Difficulty in\nRandom Constraint Satisfaction Problems\nFlorent Krz ֒aka la1 and Lenka Zdeborov ́a2\n1 PCT, UMR Gulliver 7083 CNRS-ESPCI, 10 rue Vauquelin, 75231 Paris, France\n2 LPTMS, UMR 8626 CNRS et Univ. Paris-Sud, 91405 Orsay CEDEX, France\nE-mail: fk@espci.fr\nAbstract.\nWe review the understanding of the random constraint satisfaction problems,\nfocusing on the q-coloring of large random graphs, that has been achieved using the cavity\nmethod. We also discuss the properties of the phase diagram in temperature, the connections\nwith the glass transition phenomenology in physics, and the related algorithmic issues.\n1. Introduction\nSpin glass theory has a large and probably initially unexpected impact on some problems far\nfrom condensed matter physics and one example of such spectacular outcome is the application of\nstatistical physics ideas to combinatorial optimization [1] and of the concept of phase transitions\nto the probabilistic analysis of Constraint Satisfaction Problems (CSPs) [2, 3, 4]. Given a set\nof N discrete variables subject to a set of M constraints, a CSP consists in deciding if there\nexists an assignment of the variables satisfying all the constraints. This is a generic setting that\nis currently used to tackle problems as diverse as, among others, error-correcting codes, register\nallocation in compilers or genetic regulatory networks. The class of NP-complete problems [1],\nfor which no algorithm is known that guarantees to decide satisfiability in a time polynomial in\nN, is particularly interesting. Well-studied examples of such problems are the satisfiability of\nboolean formulas (SAT), and the q-coloring problem (q-COL, see figure 1) that we shall discuss\nhere. Given a graph with N vertices and M edges connecting certain pairs of them, and given\nq colors, can we color the vertices so that no two connected vertices have the same color?\nCrucial empirical observations were made when considering the ensemble of random graphs\nwith a given average vertex connectivity c: while below a critical value cs a proper q-coloring\nof the graph exists with a probability going to one in the large size limit, it was found that\nbeyond cs no proper q-coloring exists asymptotically. This sharp threshold (which appears in\nother CSPs such as K-SAT and whose existence is partially proved in [5]) is an example of a\nphase transition arising in random CSPs. It was also observed empirically [6, 7] that deciding\ncolorability becomes on average much harder near to the coloring threshold cs than far away\nfrom it. It is therefore natural to ask ourselves: Can the value of the colorable/uncolorable\n(COL/UNCOL) phase transition be computed? Can the number of all possible colorings be also\ncomputed? Are there other interesting phase transitions? Can these transitions explain the fact\nthat solutions are sometimes very hard to find? Can this knowledge help us in designing new\nalgorithms? These questions, and their answers, are at the roots of the interest of the statistical\nphysics community in optimization problems [3, 4]."},{"paragraph_id":"p2","order":2,"text":"2. A Potts anti-ferromagnet on random graphs\nIt is immediate to realize that the q-coloring problem is equivalent to the question of determining\nif the ground-state energy of a Potts anti-ferromagnet on a random graph is zero or not [8].\nConsider indeed a graph G = (V, E) defined by its vertices V = {1, . . . , N} and edges (i, j) ∈E\nwhich connect pairs of vertices i, j ∈V; and the Hamiltonian\nH({s}) =\nX\n(i,j)∈E\nδ(si, sj) .\n(1)\nWith this choice there is no energy contribution for neighbors with different colors,\nbut a positive contribution otherwise.\nThe ground state energy (the energy at zero\ntemperature) is thus zero if and only if the graph is q-colorable.\nThis transforms the\ncoloring problem into a well-defined statistical physics model. Usually, two types of random\ngraphs are considered:\nin the c−regular ensemble all points are connected to exactly\nc neighbors, while in the Erd ̋os-R ́enyi case the connectivity has a Poisson distribution.\nFigure 1. Example: a proper 3-\ncoloring of a small graph.\n3. Cavity method: Warnings, Beliefs and Surveys\nOver the last few years,\na number of studies have\ninvestigated CSPs following the adaptation of the so-called\ncavity method [2] to random graphs [4, 9]. It is a powerful\nheuristic tool —whose exactness is widely accepted but\nhas still to be rigorously demonstrated— equivalent to the\nreplica method of disordered systems [2].\nIts main idea\nlies in the fact that a large random graph is locally tree-\nlike, and that an iterative procedure known in physics as\nthe Bethe-Peirls method can solve exactly any model on\na tree (such models are often qualified as “mean field”\nin physics).\nInterestingly, it was realized [10] that an\nequivalent formalism has been developed independently in\ncomputer science [11], where it is called Belief Propagation\n(BP, which conveniently enough, may also stands for\nBethe-Peirls). Defining ψi→j\nc\n(c = 1, .., q) as the probability that the spin i has color c in absence\nof the spin j (the “belief” that the spin j has on the properties of the spin i), BP reads\nψi→j\nc\n=\n1\nZi→j\n0\nY\nk∈N(i)\\{j}"},{"paragraph_id":"p3","order":3,"text":"1 −ψk→i\nc"},{"paragraph_id":"p4","order":4,"text":"(2)\nwhere Zi→j\n0\nis a normalization constant and the notation k ∈N(i) \\ {j} means the set of\nneighbors of i except j. From a fixed point of these equations, the complete beliefs in presence\nof all spins can be also computed. They give, for each vertex, the probability of each color from\nwhich other quantities, as for instance the number of solutions, can be computed. A simpler\nformalism, called Warning Propagation (WP), restricts itself to frozen variables (i.e. to variables\nfor which only one color can satisfy the constraints). However, WP does not allow to compute\nthe number of solutions, only their existence, but is definitely simpler to handle.\nIt was soon realized, however, that these methods developed for trees could not be\nused straightforwardly on all random graphs because of a non-trivial phenomenon called\nclustering [9, 12] (for which rigorous results are now available, see [13]).\nIndeed, while for\ngraphs with very low connectivities all solutions are “connected” —in the sense that it is easy\nwith a local dynamics to move from one solution to another— they regroup into a large number"},{"paragraph_id":"p5","order":5,"text":"of disconnected clusters for larger connectivities. It can be argued that each of these clusters\ncorresponds to a different fixed point of the BP equations, so that a survey over the whole set of\nthe fixed points should be performed. This can be done in the cavity method by the now famous\nSurvey Propagation (SP) equations [4] which, in the physics language, correspond to the Parisi’s\none-step Replica Symmetry Breaking (RSB) scheme [2]. Within this formalism, the number of\nclusters (which behaves as N = eNΣ, where Σ is a fundamental quantity called the complexity)\nand their sizes (the number of proper colorings inside the cluster) can be determined.\nThis formalism has been applied on the SAT [4] and COL [14, 15] problems in the limit\nof infinitely large graphs. These cutting edge studies were however restricted to SP applied\nto the clusters corresponding to fixed points of WP and not to those of BP. Although this\nalready allowed the correct computation of the COL/UNCOL transition and the development\nof a powerful algorithm [4], it meant that the description of the clustered phase was only partial\nand this resulted in a number of problems and inconsistencies that stayed unanswered until very\nrecently. These issues have been today clarified [16, 17, 18, 19, 20, 21] and we shall now discuss\nthis new understanding.\n4. The phase diagram of the coloring problem on a random graph\nConsider that we have q ≥4 colors (the q = 3 case being a bit particular [16, 18, 20], as\nwe shall see) and a large random graph whose connectivity c we shall increase.\nDifferent\nphases are encountered that we will now describe (and enumerate) in order of appearance (the\ncorresponding phase diagram is depicted in figure 2).\n(i) A unique cluster exists: For low enough connectivities, all the proper colorings are\nfound in a single cluster, where it is easy to “move” from one solution to another. Only\none possible —and trivial— fixed point of the BP equations exists at this stage (as can be\nproved rigorously in some cases [23]). The entropy can be computed and reads in the large\ngraph size N limit\ns = log Nsol\nN\n= log q + c\n2 log"},{"paragraph_id":"p6","order":6,"text":"1 −1\nq"},{"paragraph_id":"p7","order":7,"text":".\n(3)\n(ii) Some (irrelevant) clusters appear: As the connectivity is slightly increased, the phase\nspace of solutions decomposes into an large (exponential) number of different clusters. It\nis tempting to identify that as the clustering transition, but it happens that all (but one)\nof these clusters contain relatively very few solutions —as compare to whole set— and that\nalmost all proper colorings still belong to one single giant cluster. Clearly, this is not a\nproper clustering phenomenon and in fact, for all practical purpose, there is still only one\nsingle cluster. equation (3) still gives the correct entropy at this stage.\n(iii) The clustered phase: For larger connectivities, the large single cluster also decomposes\ninto an exponential number of smaller ones: this now defines the genuine clustering threshold\ncd1. Beyond this threshold, a local algorithm that tries to move in the space of solutions\nwill remain prisoner of a cluster of solutions [24]. Interestingly, it can be shown that the\ntotal number of solutions is still given by equation (3) in this phase. This is because, as\nis well known in the replica method, the free energy has no singularity at the dynamical\ntransition (which is therefore not a true transition in the sense of Ehrenfest, but rather a\ndynamical or geometrical transition in the space of solutions).\n(iv) The condensed phase: As the connectivity is further increased, a new sharp phase\ntransition arises at the condensation threshold cc where most of the solutions are found in\n1 It is important to point out that the location of the clustering transitions was therefore not computed correctly\nwhen the dependence on the size of clusters was not taken into account. Also, different results were obtained\npreviously depending on whether or not unfrozen fixed points were explicitly considered."},{"paragraph_id":"p8","order":8,"text":"Cluster without frozen variable\nCluster with frozen variables\nCONDENSATION\nRIGIDITY\nUNCOL\nconnectivity\nUncolorable phase\nColorable phase\nCLUSTERING\nc\nc\nc\nc\nc\ns\nr\nd\n(i)\n(ii)\n(iii)\n(iv)\n(v)\n(vi)\nFigure 2. Sketch of the space of solutions —colored points in this representation— in the\nq-coloring problem on random graphs when the connectivity c is increased. (i) At low c, all\nsolutions belong to a single cluster. (ii) For larger c, other clusters of solutions appear but a\ngiant cluster still contains almost all solutions. (iii) At the clustering transition cd, it splits\ninto an exponentially large number of clusters. (iv) At the condensation transition cc, most\ncolorings are found in the few largest of them. (v) The rigidity transition cr (cr < cc and cr > cc\nare both possible depending on q) arises when typical solutions belong to clusters with frozen\nvariables (that are allowed only one color in the cluster). (vi) No proper coloring exists beyond\nthe COL/UNCOL threshold cs.\na finite number of the largest clusters. From this point, equation (3) is not valid anymore\nand becomes just an upper bound. The entropy is non-analytic at cc therefore this is a\ngenuine static phase transition.\n(v) The rigid phase: As mentioned in section 3, two different types of clusters exist: In the\nfirst type, that we shall call the unfrozen ones, all spins can take at least two different colors.\nIn the second type, however, a finite fraction of spins is allowed only one color within the\ncluster and are thus “frozen” into this color. These frozen clusters actually correspond to\nnon-trivial fixed points of BP and WP, while the first kind are non-trivial fixed points of BP\nonly. It follows that a transition exists, that we call rigidity, when frozen variables appear\ninside the dominant clusters (those that contains most colorings). If one takes a proper\ncoloring at random beyond cr, it will belong to a cluster where a finite fraction of variables\nis frozen into the same color. Depending on the value of q, this transition may arise before\nor after the condensation transition (see table 1).\n(vi) The UNCOL phase: Eventually, the connectivity cs is reached beyond which no more\nsolutions exist. The ground state energy (sketched in figure 3) is zero for c < cs and then\ngrows continuously for c > cs. The values cs computed within the cavity formalism are in\nperfect agreement with the rigorous bounds [25] derived using probabilistic methods and\nare widely believed to be exact (although they remains to be rigorously proven, but see [26]\nfor a proof that they are at least rigorous upper bounds).\nWe report the values of the threshold connectivities corresponding to all these transitions in\ntable 1 for the regular and the Poissonian (i.e. Erd ̋os-R ́enyi) random graphs ensembles. Notice\nthat the 3-coloring is peculiar because cd = cc so that the clustered phase is always condensed in\nthis case. In view of this rich phase diagram, it is important to get an intuition on the meaning\nand the properties of these different phases and, in this respect, it is interesting before entering\nthe algorithmic implications to discuss the analogies with the glass transition."},{"paragraph_id":"p9","order":9,"text":"q\ncd\ncr\ncc\ncs\n3\n5+\n-\n6\n6\n4\n9\n-\n10\n10\n5\n14\n14\n14\n15\n6\n18\n19\n19\n20\n7\n23\n-\n25\n25\n8\n29\n30\n31\n31\n9\n34\n36\n37\n37\n10\n39\n42\n43\n44\nq\ncd\ncr\ncc\ncs\n3\n4\n4.66(1)\n4\n4.687(2)\n4\n8.353(3)\n8.83(2)\n8.46(1)\n8.901(2)\n5\n12.837(3)\n13.55(2)\n13.23(1)\n13.669(2)\n6\n17.645(5)\n18.68(2)\n18.44(1)\n18.880(2)\n7\n22.705(5)\n24.16(2)\n24.01(1)\n24.455(5)\n8\n27.95(5)\n29.93(3)\n29.90(1)\n30.335(5)\n9\n33.45(5)\n35.658\n36.08(5)\n36.490(5)\n10\n39.0(1)\n41.508\n42.50(5)\n42.93(1)\nTable\n1.\nThreshold\nconnectivities\ncd\n(dynamical/clustering)\n[20,\n17,\n18,\n27],\ncr\n(rigidity/freezing) [20, 21], cc (condensation/Kauzmann) [20, 18] and cs (COL/UNCOL) [14, 15]\nfor regular (left) and Erd ̋os-R ́enyi (right) random graphs. In the large q-limit, one finds in both\ncases that [20, 18]: cr = q[log q + log log q + 1 + o(1)], cc = 2q log q −log q −2 log 2 + o(1) and\n[15]: cs = 2q log q −log q −1 + o(1).\n5. A detour into the ideal glass transition phenomenology\nTo those familiar with the replica theory and the mean field theory of glasses, the phenomenology\ndepicted in the former section should look familiar: these successive transitions are indeed very\nwell known in the picture of the ideal glass transition [28].\nThis striking analogy is in fact\nquite natural since, despite the fact that there is no disorder in the interactions in Hamiltonian\n(1), the frustration due to the loops in the random graph makes the model behaving like a\ndisordered “anti-ferromagnetic” Potts spin glass [29] and such models are known to display the\nglassy phenomenology [8, 28].\nThe phase diagram obtained on Poissonian random graphs with average connectivity c for\nq ≥4 is sketched in figure 3 (the q = 3 model is slightly different as, again, Td = Tc). At\nhigh temperature the system behaves as a liquid (or a paramagnet in the language of magnetic\nsystems). Below a temperature Td a first transition —called “dynamical”— happens and the\nsystem falls out of equilibrium.\nFor T < Td it is not possible for a physical dynamics to\nequilibrate the system and the ergodicity is broken: this is due to the appearance of exponentially\nmany different states.\nHowever, the would-be equilibrium properties of the problem remain\nsimilar (and in particular, the free-energy has no singularity at this temperature).\nOnly at\ntemperature Tc the free energy is non analytic and a true “static” glass transition happens,\ncalled the Kauzmann transition [28]. In this phase only a finite number of states does matter\nat a given connectivity. Finally, for larger connectivities, a third phenomenon is observed as\nthe temperature is further lowered, called the Gardner transition [30, 31]. It is a transition\ntowards a more complicated phase, similar to the one found in the celebrated solution of the\nSherrington-Kirkpatrick model [2]. The fact that the Gardner transition arises for connectivities\nlarger the COL/UNCOL one is very important in this respect: it shows that the study of this\nphase, that requires a more involved cavity formalism (and probably further RSB), does not\nseem to be needed in the colorable phase2. We also now recognize that the “clustering” and the\n“condensation” transitions in the coloring problem are just the zero temperature relics of the\ndynamic and Kauzmann transitions at finite temperatures.\nA similar connection with the physics of glassy system can also be obtained directly at zero\n2 The expert reader might find this puzzling, as many papers stated that the simple “one replica symmetry\nbreaking” [4, 9] solution was unstable towards a more complex solution in some region of the COL/SAT phase\n[32, 15]. However, these results were obtained neglecting the role of the sizes of the clusters: while in some cases\nmost clusters are indeed unstable, our studies [29] indicate that the relevant ones seem always stable in the COL\nphase (although the cases of 3−COL and 3−SAT might be problematic, see [20, 29])."},{"paragraph_id":"p10","order":10,"text":"d\nC\nCG\ne (c)\nGS\nCc\nCs\nEnergy\nConnectivity\nTemperature\nTd\nc\nT\nG\nT\nGardner phase\nLiquid\n\"Dynamic\"\n\"Static\" glass\nglass\nFigure\n3.\nSketch of the\nphase diagram in the color-\ning problem at finite temper-\nature (from [29]). At Td, the\nsystem falls out of equilib-\nrium (“dynamic” transition).\nAt Tc the system undergoes a\n“static” glass transition.\nFi-\nnally, at TG, a Gardner tran-\nsition appears. eGS represents\nthe ground state energy.\ntemperature via the jamming phenomenology [19] where the density of constraints (in this case\nthe volume of some non-overlapping spheres in a box of fixed volume) are increased and where\na dynamical transition is first met while some authorized configurations exist much beyond\nthis point (see for instance [33] and references therein). We thus see that the coloring problem\non random graphs translates into a very general mean field model of a complex liquid. This\nconvergence of interest between different disciplines is quite interesting in itself and allows to\ndiscuss a number of matters, as we shall now see.\n6. Onset of hardness for local search algorithms\nThe properties of the phase diagram we just discussed are based on analytical computations\nthrough the cavity method. We would like to discuss now what are the implications of these\ndifferent phases on the performance of simple local search algorithms that try to find a solution.\nThis is, however, a much harder subject to handle analytically and we shall thus leave the field of\nanalytical computations to enter the one of phenomenology. Still, the behavior of an algorithm\ntrying to find a solution is reminiscent of the behavior of the physical dynamics in glassy systems,\nand we can at least exploit this analogy in order to get an intuition for the problem that we can\nlater confirm with numerical simulations.\nIt is first tempting to identify the point cd, where a physical Monte-Carlo dynamics gets\ntrapped into a cluster, with the onset of computational hardness3. However, a second moment\nthough indicates that this should not be the case: In the glass transition phenomenology, it\nis well known that, although the system falls out of equilibrium beyond Td, its energy can be\nfurther lowered by lowering the temperature, or just by waiting a bit longer [19]. In short: the\nfact that dynamics is prisoner of a given region of the set of possible configurations does not\nmean that no solution can be found in this region. Although cd is indeed a sharp transition for\nthe Monte-Carlo sampling, there is no reason, a priori, to experience difficulties if one just want\nto find one solution beyond this point. This is particularly transparent in the analysis and the\nalgorithm introduced in [19] (and directly inspired from the analogy with jamming [33]):\n(i) Start with a graph of connectivity c and a proper coloring.\n(ii) Increase the density of constraint by adding a link in the graph.\n(iii) Use a simple algorithm in order to solve the contradiction introduced by the link. When it\nis done, go back to step (i).\nBy applying this strategy, starting from scratch (i.e. from a graph with N vertices and no\nlink), the set of all proper colorings undergoes the successive transitions described in figure 2 as\n3 When the clustering phenomena was discovered in CSPs, it was indeed initially conjectured to be responsible\nfor the onset of hardness for local search strategies [12, 4] and the clustered phase was named the “hard phase”.\nHowever, some local algorithms were found to easily beat the threshold (see [34] for SAT and [20] for COL)."},{"paragraph_id":"p11","order":11,"text":"Zero energy states\nPositive energy states\nFigure\n4.\nArtist’s view of the energy\nlandscape for low connectivities c > cd:\na\nregion dominated by canyons that reach the\nground-states.\nZero energy states\nPositive energy states\nFigure\n5.\nArtist’s view of the energy\nlandscape for large connectivities c > cd: a\nregion dominated by high mountains and deep\nvalleys.\nconnectivity increases. When the dynamical transition is reached, one is trapped inside a cluster\nof solutions, but this is not really a problem as one is still free to move inside the cluster. As\nmore links are added, the cluster size is decreasing continuously but while it still exists, the local\nalgorithm should be in principle able to find solutions nearby. Only for larger connectivities,\nwhen the cluster gets frozen, it disappears and consequently the algorithm stops. It was shown\nin Ref. [19] through numerical simulations that this strategy, using the Walk-COL [20] algorithm\nfor step (iii), is indeed efficient, and linear in N, much beyond the dynamical transition.\nThe reason why this recursive strategy becomes inefficient when the cluster in which the\ndynamics is trapped freezes is the following: if a link is put between two vertices frozen in the\nsame color, it is impossible to satisfy the constraints while remaining in the cluster. As opposed\nto the unfrozen clusters, the frozen clusters thus have a finite probability to disappear when a\nnew link is added. A cavity-like analysis [21], confirmed by numerical data [19], actually shows\nthat the number of changes that the algorithm must perform, in order to solve the contradictions\nimposed by the addition of new links, increases with the connectivity and diverges when the\nfrozen variables appear. The source of difficulties is therefore not the clustering phenomenon in\nitself, but rather the appearance of frozen variables. This makes the analysis and the prediction\nof a EASY/HARD threshold much harder since (as one can see on figure 2) clusters of different\nsizes freeze at different connectivities, although a connectivity c∗≥cr exists where all clusters\nare frozen, thus putting a strict bound to the efficiency of this procedure.\nInterestingly, even non-incremental algorithms may also pass the cd threshold [34, 20], and\nthat might come as a surprise for those having in mind the “rugged” many-valley energy\nlandscape picture of spin glasses.\nThis apparent paradox can be clarified by the following\nconsiderations: It is possible that at lower connectivity c > cd the energy landscape is dominated\nby deep canyons (figure 4), where it is in principle easy to go down as one has just to jump ahead!\nAt larger connectivities a more rugged region with many deep valleys and high mountains is\nfound (figure 5) in which case, as any mountain-hiker will undoubtly know, it takes some time to\ngo to the deepest valley because many hills have to be climbed first. This difference in behavior\nmight explain the “unreasonable efficiency” of local algorithm [7] and the performance of the\nannealing procedure beyond cd [27].\nTo further illustrate this point, consider the Walk-COL algorithm introduced in [20] (and\nadapted from a similar one in SAT [34]) defined by the following procedure\n(i) Randomly choose a spin that has the same color as at least one of its neighbors."},{"paragraph_id":"p12","order":12,"text":"1e-05\n 1e-04\n 0.001\n 0.01\n 0.1\n 1000\n 10000\n 100000\n 1e+06\n 1e+07\nFraction of unsatisfied variables\nt/N\nc=8.0\nc=8.3\nc=8.4\nc=8.5\nc=8.75\nN=50 000\nN=200 000\n 1000\n 10000\n 100000\n 1e+06\n 8\n 8.5\n 9"},{"paragraph_id":"p13","order":13,"text":"τ(c)\ncd\ncc\ncr cs\nFigure 6.\nFraction of unsatis-\nfied variables versus the number\nof attempted flips of the Walk-\nCOL algorithm divided by the\nsize of the graph in the q = 4 col-\noring: Walk-COL [20] algorithm\nis able to find some solutions in\nthe clustered phase for low c (in\nthe canyon-like region), but get\ntrapped in the high energy val-\nleys for larger c. Inset: estimated\ntime τ = t/N needed to find a\nsolution versus connectivity.\n(ii) Change randomly its color.\nAccept this change with probability one if the number of\nunsatisfied spins has been lowered, otherwise accept it with probability p (this is a parameter\nthat has to be tuned for better efficiency).\n(iii) If there are unsatisfied vertices, go to step (i) unless the maximum running time is reached.\nThis algorithm can easily find colorings for large sizes in linear time beyond cd [20], but certainly\nnot too close to the UNCOL transition where it gets trapped at higher energies (see figure 6).\nSo far, there are few analytical results about the energy landscape in this problem and it\nis likely that this will be the subject of further studies. It is unfortunately very hard to say\nfor which connectivities the landscape goes from canyons-dominated to mountains-dominated\nas this may not be a sharp transition and more a matter of —certainly algorithm-dependent—\nbasins of attraction. The rigidity transition for typical clusters is certainly a good candidate as\na crossover in behavior (as is the connectivity where all clusters are frozen).\nTo conclude, one sees that although the algorithmic issues are indeed more difficult to handle\nthan the phase diagram, at least two important points can already be made: First, the dynamical\ntransition does not correspond to the onset of hardness, and second, the source of difficulty seems\nmore to be related with the appearance of frozen variables.\n7. Message Passing and Decimation\nThe class of local search algorithms is only one part of the story.\nA different class, where\nmessages are exchanged through the nodes of the graph, was proven to be very efficient in\ncomputer science. BP and WP are examples of such procedures and it is thus interesting to\nuse the information given by the cavity analysis to discuss their performance in estimating the\nmarginals —or other informations— in the problem. A major outcome of the last years has also\nbeen the application of SP as a message passing (MP) [4].\nFrom the information given by the fixed point of the MP, an algorithm can be defined in\nthe following way [4]: (i) run the MP to obtain some information, and (ii) fix (decimate) some\nvariables according to this information. This sequence is iterated until a solution is found, or\nuntil a contradiction is met. The two parts are quite independent and each of them can be\nchanged separately (for instance a self consistent re-enforcement has been tried instead of the\ndecimation with very good results [35]).\nAccording to the cavity formalism, the iterative fixed point of BP correctly estimates the\nmarginals until the condensation at cc [18] (which, conveniently, is very close to cs). Indeed"},{"paragraph_id":"p14","order":14,"text":"the application of BP plus decimation is numerically efficient in finding solutions for both SAT\nand COL [18, 20, 22] and it would be interesting to see if this method allows to find “typical”\nsolutions beyond cd, thus bypassing the problems of Monte-Carlo algorithms. It seems that\nthe strategy is efficient even beyond cc in some cases [20], although the BP recursion does not\nalways converge (and when it does, it does it slowly) on the decimated graph, in which case\nan imprecise and approximate information is obtained. These issues are thus far from being\nproperly understood at the present time, and we hope that new works will be done in this\ndirection.\nA more powerful strategy in the clustered phase is to use SP to compute the probability\nthat a given variable is frozen is a cluster [4]. Fixing the variables which are frozen in most\nclusters seems a good way to decimate the graph. Using this method in random 3-SAT allows to\noutperform any other known algorithm and to find solutions of huge instances rapidly even very\nclose to the threshold [4, 35]. The method has been subsequently adapted for the 3−coloring in\n[14]. Interestingly, SP behaves better than BP on the decimated graph and has no problem of\nconvergence. Other strategies are possible that have not yet been exploited [20].\nThe best way to extract the informations given by the fixed point of these message passing\nprocedures is however not yet known, nor is the limit until where these strategies are efficient.\nHowever, this line of thoughts is certainly the most promising way in the direction of better\nsolving and sampling algorithms.\n8. Conclusions and perspectives\nIn this paper, we considered and reviewed some aspects of the q−coloring of large random graphs.\nWe discussed the properties of the set of solutions and the different sharp phase transitions it\nundergoes when the average connectivity is varied. The problem translates in physics into a\nmean-field model with an ideal glass transition. The cavity method has been efficient in giving\ninsight into the problem, although the important challenge of proving rigorously these results\nremains.\nIt would be interesting in this respect to confirm these predictions by performing\nextensive Monte-Carlo simulations of the phase diagram depicted in figure 3.\nWe also discussed the dynamical behavior of local search algorithms. We saw that although\nthe “dynamical” transition has a direct meaning as the point where local Monte-Carlo algorithms\nget out of equilibrium, it is not directly connected with the onset of hardness in the problem\nwhich is rather due to the fact that clusters “freeze” as the connectivity increases. So far, it has\nnot therefore been possible, despite initial hope, to have well-defined HARD and EASY phases.\nThe precise role of frozen variables and the part played by the rigidity transition, or by the\nconnectivity where all clusters becomes frozen, will undoubtly be the subject of new research,\nboth in numerical and theoretical directions, that will help to clarify these issues. A refined\nknowledge of the energy landscape would also be valuable.\nFinally, we also discuss the major breakthrough in the algorithmic strategy that emerged\nfrom the application of the cavity solution on a single given graph. It is not clear at the present\ntime what it the best way to use this approach and how efficient it can be, nor if it is possible to\ngo arbitrarily close to the satisfiability/colorability threshold for any values of q, and it is likely\nthat these questions will also trigger a lot of work in the future.\nAcknowledgments\nWe thank J. Kurchan, A. Montanari, F. Ricci-Tersenghi and G. Semerjian for the collaborations\nthat led to a substantial part of our results. We also greatly benefit from discussions with M.\nM ́ezard and R. Zecchina. We thank T. J ̈org and A. Hartmann for a critical lecture of the paper."},{"paragraph_id":"p15","order":15,"text":"References\n[1] Cook S 1971 Proc. of the 3rd Annual ACM Symp. on Theory of Computing (ACM, New York) pp. 151-\n158. Papadimitrious C H 1994 Computational Complexity (Addison-Wesley, Reading, MA). Garey M R\nand Johnson D S 1979 Computers and Intractability (Freeman, San Francisco). Dubois O, Monasson R,\nSelman B, and Zecchina R 2001 special issue of Theor. Comp. Sci. 265. 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Achlioptas D, Naor A and Peres Y 2005 Nature 435 759.\n[26] Franz S and Leone M 2003 J. Stat. Phys. 111 535.\n[27] van Mourik J and Saad D 2002 Phys. Rev. E 66 056120.\n[28] Gross D J and M ́ezard M 1984 Nuclear Physics B 240 4 431. Kirkpatrick T, Thirumalai D, and Wolynes P\n1989 Phys. Rev. A 40 1045. M ́ezard M and Parisi G 1999 Phys. Rev. Lett. 82 747.\n[29] Krzakala F and Zdeborov ́a L 2007 Potts Glass on Random Graphs Preprint arXiv:0710.3336.\n[30] Gardner E 1985 Nuclear Physics B 257 747.\n[31] Leuzzi L and Parisi G 2001 J. Stat. Phys. 103 679. Crisanti A, Leuzzi L and Parisi G 2002 J. Phys. A:\nMath. Gen. 35 481.\n[32] Montanari A, Parisi G and Ricci-Tersenghi F 2004 J. Phys. A: Math. Gen. 37 2073. Mertens S, M ́ezard M\nand Zecchina R 2003 Random Struct. Algorithms 28 3 340.\n[33] Liu A and Nagel S 1998 Nature 396 21. O’Hern C S, Silbert L E, Liu A and Nagel S 2003 Phys. Rev. E 68\n011306. Wyart M 2005 Annales de Physique 30 3.\n[34] Ardelius J and Aurell E 2006 Phys. Rev. E 74 037702.\n[35] Chavas J, Furtlehner C, M ́ezard M, and Zecchina R 2005 J. Stat. Mech. P11016."}],"pages":[{"page":1,"text":"arXiv:0711.0110v1 [cs.CC] 1 Nov 2007\nProceedings of the International Workshop on Statistical-Mechanical Informatics\nSeptember 16–19, 2007, Kyoto, Japan\nPhase Transitions and Computational Difficulty in\nRandom Constraint Satisfaction Problems\nFlorent Krz ֒aka la1 and Lenka Zdeborov ́a2\n1 PCT, UMR Gulliver 7083 CNRS-ESPCI, 10 rue Vauquelin, 75231 Paris, France\n2 LPTMS, UMR 8626 CNRS et Univ. Paris-Sud, 91405 Orsay CEDEX, France\nE-mail: fk@espci.fr\nAbstract.\nWe review the understanding of the random constraint satisfaction problems,\nfocusing on the q-coloring of large random graphs, that has been achieved using the cavity\nmethod. We also discuss the properties of the phase diagram in temperature, the connections\nwith the glass transition phenomenology in physics, and the related algorithmic issues.\n1. Introduction\nSpin glass theory has a large and probably initially unexpected impact on some problems far\nfrom condensed matter physics and one example of such spectacular outcome is the application of\nstatistical physics ideas to combinatorial optimization [1] and of the concept of phase transitions\nto the probabilistic analysis of Constraint Satisfaction Problems (CSPs) [2, 3, 4]. Given a set\nof N discrete variables subject to a set of M constraints, a CSP consists in deciding if there\nexists an assignment of the variables satisfying all the constraints. This is a generic setting that\nis currently used to tackle problems as diverse as, among others, error-correcting codes, register\nallocation in compilers or genetic regulatory networks. The class of NP-complete problems [1],\nfor which no algorithm is known that guarantees to decide satisfiability in a time polynomial in\nN, is particularly interesting. Well-studied examples of such problems are the satisfiability of\nboolean formulas (SAT), and the q-coloring problem (q-COL, see figure 1) that we shall discuss\nhere. Given a graph with N vertices and M edges connecting certain pairs of them, and given\nq colors, can we color the vertices so that no two connected vertices have the same color?\nCrucial empirical observations were made when considering the ensemble of random graphs\nwith a given average vertex connectivity c: while below a critical value cs a proper q-coloring\nof the graph exists with a probability going to one in the large size limit, it was found that\nbeyond cs no proper q-coloring exists asymptotically. This sharp threshold (which appears in\nother CSPs such as K-SAT and whose existence is partially proved in [5]) is an example of a\nphase transition arising in random CSPs. It was also observed empirically [6, 7] that deciding\ncolorability becomes on average much harder near to the coloring threshold cs than far away\nfrom it. It is therefore natural to ask ourselves: Can the value of the colorable/uncolorable\n(COL/UNCOL) phase transition be computed? Can the number of all possible colorings be also\ncomputed? Are there other interesting phase transitions? Can these transitions explain the fact\nthat solutions are sometimes very hard to find? Can this knowledge help us in designing new\nalgorithms? These questions, and their answers, are at the roots of the interest of the statistical\nphysics community in optimization problems [3, 4]."},{"page":2,"text":"2. A Potts anti-ferromagnet on random graphs\nIt is immediate to realize that the q-coloring problem is equivalent to the question of determining\nif the ground-state energy of a Potts anti-ferromagnet on a random graph is zero or not [8].\nConsider indeed a graph G = (V, E) defined by its vertices V = {1, . . . , N} and edges (i, j) ∈E\nwhich connect pairs of vertices i, j ∈V; and the Hamiltonian\nH({s}) =\nX\n(i,j)∈E\nδ(si, sj) .\n(1)\nWith this choice there is no energy contribution for neighbors with different colors,\nbut a positive contribution otherwise.\nThe ground state energy (the energy at zero\ntemperature) is thus zero if and only if the graph is q-colorable.\nThis transforms the\ncoloring problem into a well-defined statistical physics model. Usually, two types of random\ngraphs are considered:\nin the c−regular ensemble all points are connected to exactly\nc neighbors, while in the Erd ̋os-R ́enyi case the connectivity has a Poisson distribution.\nFigure 1. Example: a proper 3-\ncoloring of a small graph.\n3. Cavity method: Warnings, Beliefs and Surveys\nOver the last few years,\na number of studies have\ninvestigated CSPs following the adaptation of the so-called\ncavity method [2] to random graphs [4, 9]. It is a powerful\nheuristic tool —whose exactness is widely accepted but\nhas still to be rigorously demonstrated— equivalent to the\nreplica method of disordered systems [2].\nIts main idea\nlies in the fact that a large random graph is locally tree-\nlike, and that an iterative procedure known in physics as\nthe Bethe-Peirls method can solve exactly any model on\na tree (such models are often qualified as “mean field”\nin physics).\nInterestingly, it was realized [10] that an\nequivalent formalism has been developed independently in\ncomputer science [11], where it is called Belief Propagation\n(BP, which conveniently enough, may also stands for\nBethe-Peirls). Defining ψi→j\nc\n(c = 1, .., q) as the probability that the spin i has color c in absence\nof the spin j (the “belief” that the spin j has on the properties of the spin i), BP reads\nψi→j\nc\n=\n1\nZi→j\n0\nY\nk∈N(i)\\{j}\n \n1 −ψk→i\nc\n \n(2)\nwhere Zi→j\n0\nis a normalization constant and the notation k ∈N(i) \\ {j} means the set of\nneighbors of i except j. From a fixed point of these equations, the complete beliefs in presence\nof all spins can be also computed. They give, for each vertex, the probability of each color from\nwhich other quantities, as for instance the number of solutions, can be computed. A simpler\nformalism, called Warning Propagation (WP), restricts itself to frozen variables (i.e. to variables\nfor which only one color can satisfy the constraints). However, WP does not allow to compute\nthe number of solutions, only their existence, but is definitely simpler to handle.\nIt was soon realized, however, that these methods developed for trees could not be\nused straightforwardly on all random graphs because of a non-trivial phenomenon called\nclustering [9, 12] (for which rigorous results are now available, see [13]).\nIndeed, while for\ngraphs with very low connectivities all solutions are “connected” —in the sense that it is easy\nwith a local dynamics to move from one solution to another— they regroup into a large number"},{"page":3,"text":"of disconnected clusters for larger connectivities. It can be argued that each of these clusters\ncorresponds to a different fixed point of the BP equations, so that a survey over the whole set of\nthe fixed points should be performed. This can be done in the cavity method by the now famous\nSurvey Propagation (SP) equations [4] which, in the physics language, correspond to the Parisi’s\none-step Replica Symmetry Breaking (RSB) scheme [2]. Within this formalism, the number of\nclusters (which behaves as N = eNΣ, where Σ is a fundamental quantity called the complexity)\nand their sizes (the number of proper colorings inside the cluster) can be determined.\nThis formalism has been applied on the SAT [4] and COL [14, 15] problems in the limit\nof infinitely large graphs. These cutting edge studies were however restricted to SP applied\nto the clusters corresponding to fixed points of WP and not to those of BP. Although this\nalready allowed the correct computation of the COL/UNCOL transition and the development\nof a powerful algorithm [4], it meant that the description of the clustered phase was only partial\nand this resulted in a number of problems and inconsistencies that stayed unanswered until very\nrecently. These issues have been today clarified [16, 17, 18, 19, 20, 21] and we shall now discuss\nthis new understanding.\n4. The phase diagram of the coloring problem on a random graph\nConsider that we have q ≥4 colors (the q = 3 case being a bit particular [16, 18, 20], as\nwe shall see) and a large random graph whose connectivity c we shall increase.\nDifferent\nphases are encountered that we will now describe (and enumerate) in order of appearance (the\ncorresponding phase diagram is depicted in figure 2).\n(i) A unique cluster exists: For low enough connectivities, all the proper colorings are\nfound in a single cluster, where it is easy to “move” from one solution to another. Only\none possible —and trivial— fixed point of the BP equations exists at this stage (as can be\nproved rigorously in some cases [23]). The entropy can be computed and reads in the large\ngraph size N limit\ns = log Nsol\nN\n= log q + c\n2 log\n \n1 −1\nq\n \n.\n(3)\n(ii) Some (irrelevant) clusters appear: As the connectivity is slightly increased, the phase\nspace of solutions decomposes into an large (exponential) number of different clusters. It\nis tempting to identify that as the clustering transition, but it happens that all (but one)\nof these clusters contain relatively very few solutions —as compare to whole set— and that\nalmost all proper colorings still belong to one single giant cluster. Clearly, this is not a\nproper clustering phenomenon and in fact, for all practical purpose, there is still only one\nsingle cluster. equation (3) still gives the correct entropy at this stage.\n(iii) The clustered phase: For larger connectivities, the large single cluster also decomposes\ninto an exponential number of smaller ones: this now defines the genuine clustering threshold\ncd1. Beyond this threshold, a local algorithm that tries to move in the space of solutions\nwill remain prisoner of a cluster of solutions [24]. Interestingly, it can be shown that the\ntotal number of solutions is still given by equation (3) in this phase. This is because, as\nis well known in the replica method, the free energy has no singularity at the dynamical\ntransition (which is therefore not a true transition in the sense of Ehrenfest, but rather a\ndynamical or geometrical transition in the space of solutions).\n(iv) The condensed phase: As the connectivity is further increased, a new sharp phase\ntransition arises at the condensation threshold cc where most of the solutions are found in\n1 It is important to point out that the location of the clustering transitions was therefore not computed correctly\nwhen the dependence on the size of clusters was not taken into account. Also, different results were obtained\npreviously depending on whether or not unfrozen fixed points were explicitly considered."},{"page":4,"text":"Cluster without frozen variable\nCluster with frozen variables\nCONDENSATION\nRIGIDITY\nUNCOL\nconnectivity\nUncolorable phase\nColorable phase\nCLUSTERING\nc\nc\nc\nc\nc\ns\nr\nd\n(i)\n(ii)\n(iii)\n(iv)\n(v)\n(vi)\nFigure 2. Sketch of the space of solutions —colored points in this representation— in the\nq-coloring problem on random graphs when the connectivity c is increased. (i) At low c, all\nsolutions belong to a single cluster. (ii) For larger c, other clusters of solutions appear but a\ngiant cluster still contains almost all solutions. (iii) At the clustering transition cd, it splits\ninto an exponentially large number of clusters. (iv) At the condensation transition cc, most\ncolorings are found in the few largest of them. (v) The rigidity transition cr (cr < cc and cr > cc\nare both possible depending on q) arises when typical solutions belong to clusters with frozen\nvariables (that are allowed only one color in the cluster). (vi) No proper coloring exists beyond\nthe COL/UNCOL threshold cs.\na finite number of the largest clusters. From this point, equation (3) is not valid anymore\nand becomes just an upper bound. The entropy is non-analytic at cc therefore this is a\ngenuine static phase transition.\n(v) The rigid phase: As mentioned in section 3, two different types of clusters exist: In the\nfirst type, that we shall call the unfrozen ones, all spins can take at least two different colors.\nIn the second type, however, a finite fraction of spins is allowed only one color within the\ncluster and are thus “frozen” into this color. These frozen clusters actually correspond to\nnon-trivial fixed points of BP and WP, while the first kind are non-trivial fixed points of BP\nonly. It follows that a transition exists, that we call rigidity, when frozen variables appear\ninside the dominant clusters (those that contains most colorings). If one takes a proper\ncoloring at random beyond cr, it will belong to a cluster where a finite fraction of variables\nis frozen into the same color. Depending on the value of q, this transition may arise before\nor after the condensation transition (see table 1).\n(vi) The UNCOL phase: Eventually, the connectivity cs is reached beyond which no more\nsolutions exist. The ground state energy (sketched in figure 3) is zero for c < cs and then\ngrows continuously for c > cs. The values cs computed within the cavity formalism are in\nperfect agreement with the rigorous bounds [25] derived using probabilistic methods and\nare widely believed to be exact (although they remains to be rigorously proven, but see [26]\nfor a proof that they are at least rigorous upper bounds).\nWe report the values of the threshold connectivities corresponding to all these transitions in\ntable 1 for the regular and the Poissonian (i.e. Erd ̋os-R ́enyi) random graphs ensembles. Notice\nthat the 3-coloring is peculiar because cd = cc so that the clustered phase is always condensed in\nthis case. In view of this rich phase diagram, it is important to get an intuition on the meaning\nand the properties of these different phases and, in this respect, it is interesting before entering\nthe algorithmic implications to discuss the analogies with the glass transition."},{"page":5,"text":"q\ncd\ncr\ncc\ncs\n3\n5+\n-\n6\n6\n4\n9\n-\n10\n10\n5\n14\n14\n14\n15\n6\n18\n19\n19\n20\n7\n23\n-\n25\n25\n8\n29\n30\n31\n31\n9\n34\n36\n37\n37\n10\n39\n42\n43\n44\nq\ncd\ncr\ncc\ncs\n3\n4\n4.66(1)\n4\n4.687(2)\n4\n8.353(3)\n8.83(2)\n8.46(1)\n8.901(2)\n5\n12.837(3)\n13.55(2)\n13.23(1)\n13.669(2)\n6\n17.645(5)\n18.68(2)\n18.44(1)\n18.880(2)\n7\n22.705(5)\n24.16(2)\n24.01(1)\n24.455(5)\n8\n27.95(5)\n29.93(3)\n29.90(1)\n30.335(5)\n9\n33.45(5)\n35.658\n36.08(5)\n36.490(5)\n10\n39.0(1)\n41.508\n42.50(5)\n42.93(1)\nTable\n1.\nThreshold\nconnectivities\ncd\n(dynamical/clustering)\n[20,\n17,\n18,\n27],\ncr\n(rigidity/freezing) [20, 21], cc (condensation/Kauzmann) [20, 18] and cs (COL/UNCOL) [14, 15]\nfor regular (left) and Erd ̋os-R ́enyi (right) random graphs. In the large q-limit, one finds in both\ncases that [20, 18]: cr = q[log q + log log q + 1 + o(1)], cc = 2q log q −log q −2 log 2 + o(1) and\n[15]: cs = 2q log q −log q −1 + o(1).\n5. A detour into the ideal glass transition phenomenology\nTo those familiar with the replica theory and the mean field theory of glasses, the phenomenology\ndepicted in the former section should look familiar: these successive transitions are indeed very\nwell known in the picture of the ideal glass transition [28].\nThis striking analogy is in fact\nquite natural since, despite the fact that there is no disorder in the interactions in Hamiltonian\n(1), the frustration due to the loops in the random graph makes the model behaving like a\ndisordered “anti-ferromagnetic” Potts spin glass [29] and such models are known to display the\nglassy phenomenology [8, 28].\nThe phase diagram obtained on Poissonian random graphs with average connectivity c for\nq ≥4 is sketched in figure 3 (the q = 3 model is slightly different as, again, Td = Tc). At\nhigh temperature the system behaves as a liquid (or a paramagnet in the language of magnetic\nsystems). Below a temperature Td a first transition —called “dynamical”— happens and the\nsystem falls out of equilibrium.\nFor T < Td it is not possible for a physical dynamics to\nequilibrate the system and the ergodicity is broken: this is due to the appearance of exponentially\nmany different states.\nHowever, the would-be equilibrium properties of the problem remain\nsimilar (and in particular, the free-energy has no singularity at this temperature).\nOnly at\ntemperature Tc the free energy is non analytic and a true “static” glass transition happens,\ncalled the Kauzmann transition [28]. In this phase only a finite number of states does matter\nat a given connectivity. Finally, for larger connectivities, a third phenomenon is observed as\nthe temperature is further lowered, called the Gardner transition [30, 31]. It is a transition\ntowards a more complicated phase, similar to the one found in the celebrated solution of the\nSherrington-Kirkpatrick model [2]. The fact that the Gardner transition arises for connectivities\nlarger the COL/UNCOL one is very important in this respect: it shows that the study of this\nphase, that requires a more involved cavity formalism (and probably further RSB), does not\nseem to be needed in the colorable phase2. We also now recognize that the “clustering” and the\n“condensation” transitions in the coloring problem are just the zero temperature relics of the\ndynamic and Kauzmann transitions at finite temperatures.\nA similar connection with the physics of glassy system can also be obtained directly at zero\n2 The expert reader might find this puzzling, as many papers stated that the simple “one replica symmetry\nbreaking” [4, 9] solution was unstable towards a more complex solution in some region of the COL/SAT phase\n[32, 15]. However, these results were obtained neglecting the role of the sizes of the clusters: while in some cases\nmost clusters are indeed unstable, our studies [29] indicate that the relevant ones seem always stable in the COL\nphase (although the cases of 3−COL and 3−SAT might be problematic, see [20, 29])."},{"page":6,"text":"d\nC\nCG\ne (c)\nGS\nCc\nCs\nEnergy\nConnectivity\nTemperature\nTd\nc\nT\nG\nT\nGardner phase\nLiquid\n\"Dynamic\"\n\"Static\" glass\nglass\nFigure\n3.\nSketch of the\nphase diagram in the color-\ning problem at finite temper-\nature (from [29]). At Td, the\nsystem falls out of equilib-\nrium (“dynamic” transition).\nAt Tc the system undergoes a\n“static” glass transition.\nFi-\nnally, at TG, a Gardner tran-\nsition appears. eGS represents\nthe ground state energy.\ntemperature via the jamming phenomenology [19] where the density of constraints (in this case\nthe volume of some non-overlapping spheres in a box of fixed volume) are increased and where\na dynamical transition is first met while some authorized configurations exist much beyond\nthis point (see for instance [33] and references therein). We thus see that the coloring problem\non random graphs translates into a very general mean field model of a complex liquid. This\nconvergence of interest between different disciplines is quite interesting in itself and allows to\ndiscuss a number of matters, as we shall now see.\n6. Onset of hardness for local search algorithms\nThe properties of the phase diagram we just discussed are based on analytical computations\nthrough the cavity method. We would like to discuss now what are the implications of these\ndifferent phases on the performance of simple local search algorithms that try to find a solution.\nThis is, however, a much harder subject to handle analytically and we shall thus leave the field of\nanalytical computations to enter the one of phenomenology. Still, the behavior of an algorithm\ntrying to find a solution is reminiscent of the behavior of the physical dynamics in glassy systems,\nand we can at least exploit this analogy in order to get an intuition for the problem that we can\nlater confirm with numerical simulations.\nIt is first tempting to identify the point cd, where a physical Monte-Carlo dynamics gets\ntrapped into a cluster, with the onset of computational hardness3. However, a second moment\nthough indicates that this should not be the case: In the glass transition phenomenology, it\nis well known that, although the system falls out of equilibrium beyond Td, its energy can be\nfurther lowered by lowering the temperature, or just by waiting a bit longer [19]. In short: the\nfact that dynamics is prisoner of a given region of the set of possible configurations does not\nmean that no solution can be found in this region. Although cd is indeed a sharp transition for\nthe Monte-Carlo sampling, there is no reason, a priori, to experience difficulties if one just want\nto find one solution beyond this point. This is particularly transparent in the analysis and the\nalgorithm introduced in [19] (and directly inspired from the analogy with jamming [33]):\n(i) Start with a graph of connectivity c and a proper coloring.\n(ii) Increase the density of constraint by adding a link in the graph.\n(iii) Use a simple algorithm in order to solve the contradiction introduced by the link. When it\nis done, go back to step (i).\nBy applying this strategy, starting from scratch (i.e. from a graph with N vertices and no\nlink), the set of all proper colorings undergoes the successive transitions described in figure 2 as\n3 When the clustering phenomena was discovered in CSPs, it was indeed initially conjectured to be responsible\nfor the onset of hardness for local search strategies [12, 4] and the clustered phase was named the “hard phase”.\nHowever, some local algorithms were found to easily beat the threshold (see [34] for SAT and [20] for COL)."},{"page":7,"text":"Zero energy states\nPositive energy states\nFigure\n4.\nArtist’s view of the energy\nlandscape for low connectivities c > cd:\na\nregion dominated by canyons that reach the\nground-states.\nZero energy states\nPositive energy states\nFigure\n5.\nArtist’s view of the energy\nlandscape for large connectivities c > cd: a\nregion dominated by high mountains and deep\nvalleys.\nconnectivity increases. When the dynamical transition is reached, one is trapped inside a cluster\nof solutions, but this is not really a problem as one is still free to move inside the cluster. As\nmore links are added, the cluster size is decreasing continuously but while it still exists, the local\nalgorithm should be in principle able to find solutions nearby. Only for larger connectivities,\nwhen the cluster gets frozen, it disappears and consequently the algorithm stops. It was shown\nin Ref. [19] through numerical simulations that this strategy, using the Walk-COL [20] algorithm\nfor step (iii), is indeed efficient, and linear in N, much beyond the dynamical transition.\nThe reason why this recursive strategy becomes inefficient when the cluster in which the\ndynamics is trapped freezes is the following: if a link is put between two vertices frozen in the\nsame color, it is impossible to satisfy the constraints while remaining in the cluster. As opposed\nto the unfrozen clusters, the frozen clusters thus have a finite probability to disappear when a\nnew link is added. A cavity-like analysis [21], confirmed by numerical data [19], actually shows\nthat the number of changes that the algorithm must perform, in order to solve the contradictions\nimposed by the addition of new links, increases with the connectivity and diverges when the\nfrozen variables appear. The source of difficulties is therefore not the clustering phenomenon in\nitself, but rather the appearance of frozen variables. This makes the analysis and the prediction\nof a EASY/HARD threshold much harder since (as one can see on figure 2) clusters of different\nsizes freeze at different connectivities, although a connectivity c∗≥cr exists where all clusters\nare frozen, thus putting a strict bound to the efficiency of this procedure.\nInterestingly, even non-incremental algorithms may also pass the cd threshold [34, 20], and\nthat might come as a surprise for those having in mind the “rugged” many-valley energy\nlandscape picture of spin glasses.\nThis apparent paradox can be clarified by the following\nconsiderations: It is possible that at lower connectivity c > cd the energy landscape is dominated\nby deep canyons (figure 4), where it is in principle easy to go down as one has just to jump ahead!\nAt larger connectivities a more rugged region with many deep valleys and high mountains is\nfound (figure 5) in which case, as any mountain-hiker will undoubtly know, it takes some time to\ngo to the deepest valley because many hills have to be climbed first. This difference in behavior\nmight explain the “unreasonable efficiency” of local algorithm [7] and the performance of the\nannealing procedure beyond cd [27].\nTo further illustrate this point, consider the Walk-COL algorithm introduced in [20] (and\nadapted from a similar one in SAT [34]) defined by the following procedure\n(i) Randomly choose a spin that has the same color as at least one of its neighbors."},{"page":8,"text":"1e-05\n 1e-04\n 0.001\n 0.01\n 0.1\n 1000\n 10000\n 100000\n 1e+06\n 1e+07\nFraction of unsatisfied variables\nt/N\nc=8.0\nc=8.3\nc=8.4\nc=8.5\nc=8.75\nN=50 000\nN=200 000\n 1000\n 10000\n 100000\n 1e+06\n 8\n 8.5\n 9\n \n \nτ(c)\ncd\ncc\ncr cs\nFigure 6.\nFraction of unsatis-\nfied variables versus the number\nof attempted flips of the Walk-\nCOL algorithm divided by the\nsize of the graph in the q = 4 col-\noring: Walk-COL [20] algorithm\nis able to find some solutions in\nthe clustered phase for low c (in\nthe canyon-like region), but get\ntrapped in the high energy val-\nleys for larger c. Inset: estimated\ntime τ = t/N needed to find a\nsolution versus connectivity.\n(ii) Change randomly its color.\nAccept this change with probability one if the number of\nunsatisfied spins has been lowered, otherwise accept it with probability p (this is a parameter\nthat has to be tuned for better efficiency).\n(iii) If there are unsatisfied vertices, go to step (i) unless the maximum running time is reached.\nThis algorithm can easily find colorings for large sizes in linear time beyond cd [20], but certainly\nnot too close to the UNCOL transition where it gets trapped at higher energies (see figure 6).\nSo far, there are few analytical results about the energy landscape in this problem and it\nis likely that this will be the subject of further studies. It is unfortunately very hard to say\nfor which connectivities the landscape goes from canyons-dominated to mountains-dominated\nas this may not be a sharp transition and more a matter of —certainly algorithm-dependent—\nbasins of attraction. The rigidity transition for typical clusters is certainly a good candidate as\na crossover in behavior (as is the connectivity where all clusters are frozen).\nTo conclude, one sees that although the algorithmic issues are indeed more difficult to handle\nthan the phase diagram, at least two important points can already be made: First, the dynamical\ntransition does not correspond to the onset of hardness, and second, the source of difficulty seems\nmore to be related with the appearance of frozen variables.\n7. Message Passing and Decimation\nThe class of local search algorithms is only one part of the story.\nA different class, where\nmessages are exchanged through the nodes of the graph, was proven to be very efficient in\ncomputer science. BP and WP are examples of such procedures and it is thus interesting to\nuse the information given by the cavity analysis to discuss their performance in estimating the\nmarginals —or other informations— in the problem. A major outcome of the last years has also\nbeen the application of SP as a message passing (MP) [4].\nFrom the information given by the fixed point of the MP, an algorithm can be defined in\nthe following way [4]: (i) run the MP to obtain some information, and (ii) fix (decimate) some\nvariables according to this information. This sequence is iterated until a solution is found, or\nuntil a contradiction is met. The two parts are quite independent and each of them can be\nchanged separately (for instance a self consistent re-enforcement has been tried instead of the\ndecimation with very good results [35]).\nAccording to the cavity formalism, the iterative fixed point of BP correctly estimates the\nmarginals until the condensation at cc [18] (which, conveniently, is very close to cs). Indeed"},{"page":9,"text":"the application of BP plus decimation is numerically efficient in finding solutions for both SAT\nand COL [18, 20, 22] and it would be interesting to see if this method allows to find “typical”\nsolutions beyond cd, thus bypassing the problems of Monte-Carlo algorithms. It seems that\nthe strategy is efficient even beyond cc in some cases [20], although the BP recursion does not\nalways converge (and when it does, it does it slowly) on the decimated graph, in which case\nan imprecise and approximate information is obtained. These issues are thus far from being\nproperly understood at the present time, and we hope that new works will be done in this\ndirection.\nA more powerful strategy in the clustered phase is to use SP to compute the probability\nthat a given variable is frozen is a cluster [4]. Fixing the variables which are frozen in most\nclusters seems a good way to decimate the graph. Using this method in random 3-SAT allows to\noutperform any other known algorithm and to find solutions of huge instances rapidly even very\nclose to the threshold [4, 35]. The method has been subsequently adapted for the 3−coloring in\n[14]. Interestingly, SP behaves better than BP on the decimated graph and has no problem of\nconvergence. Other strategies are possible that have not yet been exploited [20].\nThe best way to extract the informations given by the fixed point of these message passing\nprocedures is however not yet known, nor is the limit until where these strategies are efficient.\nHowever, this line of thoughts is certainly the most promising way in the direction of better\nsolving and sampling algorithms.\n8. Conclusions and perspectives\nIn this paper, we considered and reviewed some aspects of the q−coloring of large random graphs.\nWe discussed the properties of the set of solutions and the different sharp phase transitions it\nundergoes when the average connectivity is varied. The problem translates in physics into a\nmean-field model with an ideal glass transition. The cavity method has been efficient in giving\ninsight into the problem, although the important challenge of proving rigorously these results\nremains.\nIt would be interesting in this respect to confirm these predictions by performing\nextensive Monte-Carlo simulations of the phase diagram depicted in figure 3.\nWe also discussed the dynamical behavior of local search algorithms. We saw that although\nthe “dynamical” transition has a direct meaning as the point where local Monte-Carlo algorithms\nget out of equilibrium, it is not directly connected with the onset of hardness in the problem\nwhich is rather due to the fact that clusters “freeze” as the connectivity increases. So far, it has\nnot therefore been possible, despite initial hope, to have well-defined HARD and EASY phases.\nThe precise role of frozen variables and the part played by the rigidity transition, or by the\nconnectivity where all clusters becomes frozen, will undoubtly be the subject of new research,\nboth in numerical and theoretical directions, that will help to clarify these issues. A refined\nknowledge of the energy landscape would also be valuable.\nFinally, we also discuss the major breakthrough in the algorithmic strategy that emerged\nfrom the application of the cavity solution on a single given graph. It is not clear at the present\ntime what it the best way to use this approach and how efficient it can be, nor if it is possible to\ngo arbitrarily close to the satisfiability/colorability threshold for any values of q, and it is likely\nthat these questions will also trigger a lot of work in the future.\nAcknowledgments\nWe thank J. Kurchan, A. Montanari, F. Ricci-Tersenghi and G. Semerjian for the collaborations\nthat led to a substantial part of our results. We also greatly benefit from discussions with M.\nM ́ezard and R. Zecchina. We thank T. J ̈org and A. Hartmann for a critical lecture of the paper."},{"page":10,"text":"References\n[1] Cook S 1971 Proc. of the 3rd Annual ACM Symp. on Theory of Computing (ACM, New York) pp. 151-\n158. Papadimitrious C H 1994 Computational Complexity (Addison-Wesley, Reading, MA). Garey M R\nand Johnson D S 1979 Computers and Intractability (Freeman, San Francisco). Dubois O, Monasson R,\nSelman B, and Zecchina R 2001 special issue of Theor. Comp. Sci. 265. 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E 70 046705.\n[16] M ́ezard M, Palassini M and Rivoire O 2005 Phys. Rev. Lett. 95 200202.\n[17] M ́ezard M and Montanari A 2006 J. Stat. Phys. 124 1317.\n[18] Krzakala F, Montanari A, Ricci-Tersenghi F, Semerjian G and Zdeborov ́a L 2007 Proc. Natl. Acad. Sci.\n104 10318.\n[19] Krzakala F and Kurchan J 2007 Phys. Rev. E 76 021122.\n[20] Zdeborov ́a L and Krzakala F 2007 Phys. Rev. E 76 031131.\n[21] Semerjian G 2007 On the freezing of variables in random constraint satisfaction problems Preprint\narXiv:0705.2147 (J. Stat. Phys., on press).\n[22] Montanari A, Ricci-Tersenghi F, Semerjian F 2007 Solving Constraint Satisfaction Problems through Belief\nPropagation-guided decimation Preprint arXiv:0709.1667.\n[23] Bandyopadhyay A and Gamarnik D 2006 ACM-SIAM Symp. on Discrete Algorithms 890. Jonasson J 2002\nStatistics and Probability Letters 57 243.\n[24] Montanari A and Semerjian G 2006 J. Stat. Phys. 124 103.\n[25] Luczak T 1991 Combinatorica 11 45. Achlioptas D, Naor A and Peres Y 2005 Nature 435 759.\n[26] Franz S and Leone M 2003 J. Stat. Phys. 111 535.\n[27] van Mourik J and Saad D 2002 Phys. Rev. E 66 056120.\n[28] Gross D J and M ́ezard M 1984 Nuclear Physics B 240 4 431. Kirkpatrick T, Thirumalai D, and Wolynes P\n1989 Phys. Rev. A 40 1045. M ́ezard M and Parisi G 1999 Phys. Rev. Lett. 82 747.\n[29] Krzakala F and Zdeborov ́a L 2007 Potts Glass on Random Graphs Preprint arXiv:0710.3336.\n[30] Gardner E 1985 Nuclear Physics B 257 747.\n[31] Leuzzi L and Parisi G 2001 J. Stat. Phys. 103 679. Crisanti A, Leuzzi L and Parisi G 2002 J. Phys. A:\nMath. Gen. 35 481.\n[32] Montanari A, Parisi G and Ricci-Tersenghi F 2004 J. Phys. A: Math. Gen. 37 2073. Mertens S, M ́ezard M\nand Zecchina R 2003 Random Struct. Algorithms 28 3 340.\n[33] Liu A and Nagel S 1998 Nature 396 21. O’Hern C S, Silbert L E, Liu A and Nagel S 2003 Phys. Rev. E 68\n011306. Wyart M 2005 Annales de Physique 30 3.\n[34] Ardelius J and Aurell E 2006 Phys. Rev. E 74 037702.\n[35] Chavas J, Furtlehner C, M ́ezard M, and Zecchina R 2005 J. Stat. Mech. P11016."}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"Consider indeed a graph G = (V, E) defined by its vertices V = {1, . . . , N} and edges (i, j) ∈E","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"H({s}) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"(c = 1, .., q) as the probability that the spin i has color c in absence","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"clusters (which behaves as N = eNΣ, where Σ is a fundamental quantity called the complexity)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"Consider that we have q ≥4 colors (the q = 3 case being a bit particular [16, 18, 20], as","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"s = log Nsol","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"= log q + c","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"that the 3-coloring is peculiar because cd = cc so that the clustered phase is always condensed in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"cases that [20, 18]: cr = q[log q + log log q + 1 + o(1)], cc = 2q log q −log q −2 log 2 + o(1) and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"[15]: cs = 2q log q −log q −1 + o(1).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"q ≥4 is sketched in figure 3 (the q = 3 model is slightly different as, again, Td = Tc). At","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"c=8.0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"c=8.3","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"c=8.4","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"c=8.5","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"c=8.75","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"N=50 000","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"N=200 000","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"size of the graph in the q = 4 col-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"time τ = t/N needed to find a","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":35856,"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}}