Biased measures for random constraint satisfaction problems: larger interaction range and asymptotic expansion

Budzynski, Louise; Semerjian, Guilhem

doi:10.1088/1742-5468/abb8c8

Cited by 4 publications

(3 citation statements)

References 56 publications

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“…Unfortunately, though, for large K, the gain is only in second-order terms, and up to those orders, the threshold for the onset of weak clustering based on biased counting measures is still 2 K log 2=K, as shown by Budzynski and Semerjian in ref. 39. This, arguably, provides an even more robust evidence that this value marks a phase-transition point of fundamental nature.…”

Section: In Search Of the Right Algorithmic Complexity Theorymentioning

confidence: 77%

The overlap gap property: A topological barrier to optimizing over random structures

Gamarnik

2021

Proc. Natl. Acad. Sci. U.S.A.

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The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures, finding optimal solutions by means of fast algorithms is not known and often is believed not to be possible. At the same time, the formal hardness of these problems in the form of the complexity-theoretic NP-hardness is lacking. A new approach for algorithmic intractability in random structures is described in this article, which is based on the topological disconnectivity property of the set of pairwise distances of near-optimal solutions, called the Overlap Gap Property. The article demonstrates how this property 1) emerges in most models known to exhibit an apparent algorithmic hardness; 2) is consistent with the hardness/tractability phase transition for many models analyzed to the day; and, importantly, 3) allows to mathematically rigorously rule out a large class of algorithms as potential contenders, specifically the algorithms that exhibit the input stability (insensitivity).

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Section: In Search Of the Right Algorithmic Complexity Theorymentioning

confidence: 77%

The overlap gap property: A topological barrier to optimizing over random structures

Gamarnik

2021

Proc. Natl. Acad. Sci. U.S.A.

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“…Interestingly, for small values of K this can be done effectively delaying the onset of weak clustering, as was shown by Budzynski et al in [BRTS19]. Unfortunately, though, for large K the gain is only in second order terms, and up to those orders, the threshold for the onset of weak clustering based on biased counting measures is still 2 K log 2/K as shown by Budzynski and Semerjian in [BS20]. This arguably, provides an even more robust evidence that this value marks a phase transition point of fundamental nature.…”

Section: In Search Of the "Right" Algorithmic Complexity Theorymentioning

confidence: 83%

The Overlap Gap Property: a Geometric Barrier to Optimizing over Random Structures

Gamarnik

2021

Preprint

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The problem of optimizing over random structures emerges in many areas of science and engineering, ranging from statistical physics to machine learning and artificial intelligence. For many such structures finding optimal solutions by means of fast algorithms is not known and often is believed not possible. At the same time the formal hardness of these problems in form of say complexity-theoretic N P -hardness is lacking.In this introductory article a new approach for algorithmic intractability in random structures is described, which is based on the topological disconnectivity property of the set of pair-wise distances of near optimal solutions, called the Overlap Gap Property. The article demonstrates how this property a) emerges in most models known to exhibit an apparent algorithmic hardness b) is consistent with the hardness/tractability phase transition for many models analyzed to the day, and importantly c) allows to mathematically rigorously rule out large classes of algorithms as potential contenders, in particular the algorithms exhibiting the input stability (insensitivity).

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“…We will perform an optimization of the interaction potential with the aim of postponing as more as possible the dynamical phase transition. This has been done recently in [13] for the discrete CSP of bicoloring random hypergraphs (a generalization to larger interaction range for the bias is considered in [14]) and was done in [15,16] for models of hard spheres in infinite dimensions. The idea behind this optimization is that, by working with the reweighted interaction potential, the long range correlations leading to the ergodicity breaking at the dynamic phase transition will appear later, and algorithms should find solutions in an easier way in these 'biased landscapes'.…”

Section: Introductionmentioning

confidence: 99%

Optimization of the dynamic transition in the continuous coloring problem

Cavaliere¹,

Lesieur²,

Ricci-Tersenghi³

2021

J. Stat. Mech.

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Random constraint satisfaction problems (CSPs) can exhibit a phase where the number of constraints per variable α makes the system solvable in theory on the one hand, but also makes the search for a solution hard, meaning that common algorithms such as Monte Carlo (MC) method fail to find a solution. The onset of this hardness is deeply linked to the appearance of a dynamical phase transition where the phase space of the problem breaks into an exponential number of clusters. The exact position of this dynamical phase transition is not universal with respect to the details of the Hamiltonian one chooses to represent a given problem. In this paper, we develop some theoretical tools in order to find a systematic way to build a Hamiltonian that maximizes the dynamic α d threshold. To illustrate our techniques, we will concentrate on the problem of continuous coloring, where one tries to set an angle x i ∈ [0; 2π] on each node of a network in such a way that no adjacent nodes are closer than some threshold angle θ, that is cos(x i − x j )⩽ cos θ. This problem can be both seen as a continuous version of the discrete graph coloring problem or as a one-dimensional version of the Mari–Krzakala–Kurchan model. The relevance of this model stems from the fact that continuous CSPs on sparse random graphs remain largely unexplored in statistical physics. We show that for sufficiently small angle θ this model presents a random first order transition and compute the dynamical, condensation and Kesten–Stigum transitions; we also compare the analytical predictions with MC simulations for values of θ = 2π/q, q ∈ N . Choosing such values of q allows us to easily compare our results with the renowned problem of discrete coloring.

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Biased measures for random constraint satisfaction problems: larger interaction range and asymptotic expansion

Cited by 4 publications

References 56 publications

The overlap gap property: A topological barrier to optimizing over random structures

The overlap gap property: A topological barrier to optimizing over random structures

The Overlap Gap Property: a Geometric Barrier to Optimizing over Random Structures

Optimization of the dynamic transition in the continuous coloring problem

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