Biased landscapes for random Constraint Satisfaction Problems

The typical complexity of Constraint Satisfaction Problems (CSPs) can be investigated by means of random ensembles of instances. The latter exhibit many threshold phenomena besides their satisfiability phase transition, in particular a clustering or dynamic phase transition (related to the tree reconstruction problem) at which their typical solutions shatter into disconnected components. In this paper we study the evolution of this phenomenon under a bias that breaks the uniformity among solutions of one CSP instance, concentrating on the bicoloring of k-uniform random hypergraphs. We show that for small k the clustering transition can be delayed in this way to higher density of constraints, and that this strategy has a positive impact on the performances of Simulated Annealing algorithms. We characterize the modest gain that can be expected in the large k limit from the simple implementation of the biasing idea studied here. This paper contains also a contribution of a more methodological nature, made of a review and extension of the methods to determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure

From algorithms to connectivity and back: finding a giant component in random k-SAT

We take an algorithmic approach to studying the solution space geometry of relatively sparse random and bounded degree $k$-CNFs for large $k$. In the course of doing so, we establish that with high probability, a random $k$-CNF $\Phi$ with $n$ variables and clause density $\alpha = m/n \lesssim 2^{k/6}$ has a giant component of solutions that are connected in a graph where solutions are adjacent if they have Hamming distance $O_k(\log n)$ and that a similar result holds for bounded degree $k$-CNFs at similar densities. We are also able to deduce looseness results for random and bounded degree $k$-CNFs in a similar regime. Although our main motivation was understanding the geometry of the solution space, our methods have algorithmic implications. Towards that end, we construct an idealized block dynamics that samples solutions from a random $k$-CNF $\Phi$ with density $\alpha = m/n \lesssim 2^{k/52}$. We show this Markov chain can with high probability be implemented in polynomial time and by leveraging spectral independence, we also observe that it mixes relatively fast, giving a polynomial time algorithm to with high probability sample a uniformly random solution to a random $k$-CNF. Our work suggests that the natural route to pinning down when a giant component exists is to develop sharper algorithms for sampling solutions in random $k$-CNFs.Comment: 41 pages, 1 figur

Counting Solutions to Random CNF Formulas

We give the first efficient algorithm to approximately count the number of solutions in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previous counting algorithm was due to Montanari and Shah and was based on the correlation decay method, which works up to densities $(1+o_k(1))\frac{2\log k}{k}$, the Gibbs uniqueness threshold for the model. Instead, our algorithm harnesses a recent technique by Moitra to work for random formulas. The main challenge in our setting is to account for the presence of high-degree variables whose marginal distributions are hard to control and which cause significant correlations within the formula

Improved Bounds for Sampling Solutions of Random CNF Formulas

Let $\Phi$ be a random $k$-CNF formula on $n$ variables and $m$ clauses, where each clause is a disjunction of $k$ literals chosen independently and uniformly. Our goal is to sample an approximately uniform solution of $\Phi$ (or equivalently, approximate the partition function of $\Phi$). Let $\alpha=m/n$ be the density. The previous best algorithm runs in time $n^{\mathsf{poly}(k,\alpha)}$ for any $\alpha\lesssim2^{k/300}$ [Galanis, Goldberg, Guo, and Yang, SIAM J. Comput.'21]. Our result significantly improves both bounds by providing an almost-linear time sampler for any $\alpha\lesssim2^{k/3}$. The density $\alpha$ captures the \emph{average degree} in the random formula. In the worst-case model with bounded \emph{maximum degree}, current best efficient sampler works up to degree bound $2^{k/5}$ [He, Wang, and Yin, FOCS'22 and SODA'23], which is, for the first time, superseded by its average-case counterpart due to our $2^{k/3}$ bound. Our result is the first progress towards establishing the intuition that the solvability of the average-case model (random $k$-CNF formula with bounded average degree) is better than the worst-case model (standard $k$-CNF formula with bounded maximal degree) in terms of sampling solutions.Comment: 51 pages, all proofs added, and bounds slightly improve

Fast sampling of satisfying assignments from random kk-SAT

We give a nearly linear-time algorithm to approximately sample satisfying assignments in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previously known sampling algorithm for the random $k$-SAT model applies when the density $\alpha=m/n$ of the formula is less than $2^{k/300}$ and runs in time $n^{\exp(\Theta(k))}$ (Galanis, Goldberg, Guo and Yang, SIAM J. Comput., 2021). Here $n$ is the number of variables and $m$ is the number of clauses. Our algorithm achieves a significantly faster running time of $n^{1 + o_k(1)}$ and samples satisfying assignments up to density $\alpha\leq 2^{0.039 k}$. The main challenge in our setting is the presence of many variables with unbounded degree, which causes significant correlations within the formula and impedes the application of relevant Markov chain methods from the bounded-degree setting (Feng, Guo, Yin and Zhang, J. ACM, 2021; Jain, Pham and Vuong, 2021). Our main technical contribution is a $o_k(\log n )$ bound of the sum of influences in the $k$-SAT model which turns out to be robust against the presence of high-degree variables. This allows us to apply the spectral independence framework and obtain fast mixing results of a uniform-block Glauber dynamics on a carefully selected subset of the variables. The final key ingredient in our method is to take advantage of the sparsity of logarithmic-sized connected sets and the expansion properties of the random formula, and establish relevant properties of the set of satisfying assignments that enable the fast simulation of this Glauber dynamics.Comment: 47 page

Approximate counting via complex zero-free regions and spectral independence

This thesis investigates fundamental problems in approximate counting that arise in the field of statistical mechanics. Building upon recent advancements in the area, our research aims to enhance our understanding of the computational complexity of sampling from the Ising and Potts models, as well as the random $k$-SAT model. The $q$-state Potts model is a spin model in which each particle is randomly assigned a spin (out of $q$ possible spins), where the probability of a certain assignment depends on how many adjacent particles present the same spin. The edge interaction of the model is a parameter that quantifies the strength of interaction between two adjacent particles. The Ising model corresponds to the Potts model with $q = 2$. Sampling from these models is inherently connected to approximating the partition function of the model, a graph polynomial that encodes several aggregate thermodynamic properties of the system. In addition to classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane of these partition functions, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Thus, following this trend in both statistical physics and algorithmic research, we allow the edge interaction to be any complex number. First, we study the complexity of approximating the partition function of the $q$-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Previous work in the complex plane by Goldberg and Guo focused on $q=2$; for $q>2$, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing \#P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all $q\geq 2$ and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in the context of quantum computations. Secondly, we investigate the complexity of approximating the partition function \ising(G; \beta) of the Ising model in terms of the relation between the edge interaction $\beta$ and a parameter $\Delta$ which is an upper bound on the maximum degree of the input graph $G$. In this thesis we establish both new tractability and inapproximability results. Our tractability results show that \ising(-; \beta) has an FPTAS when $\beta \in \mathbb{C}$ and $\lvert \beta - 1 \rvert / \lvert \beta + 1 \rvert 1 / \sqrt{\Delta - 1}$. These are the first results to show intractability of approximating \ising(-, \beta) on bounded degree graphs with complex $\beta$. Moreover, we demonstrate situations in which zeros of the partition function imply hardness of approximation in the Ising model. Finally, we exploit the recently successful framework of spectral independence to analyse the mixing time of a Markov chain, and we apply it in order to sample satisfying assignments of $k$-CNF formulas. Our analysis leads to a nearly linear-time algorithm to approximately sample satisfying assignments in the random $k$-SAT model when the density of the random formula $\alpha=m/n$ scales exponentially with $k$, where $n$ is the number of variables and $m$ is the number of clauses. The best previously known sampling algorithm for the random $k$-SAT model applies when the density $\alpha=m/n$ of the formula is less than $2^{k/300}$ and runs in time $n^{\exp(\Theta(k))}$. Our algorithm achieves a significantly faster running time of $n^{1 + o_k(1)}$ and samples satisfying assignments up to density $\alpha\leq 2^{0.039 k}$. The main challenge in our setting is the presence of many variables with unbounded degree, which causes significant correlations within the formula and impedes the application of relevant Markov chain methods from the bounded-degree setting

Analyzing Walksat on Random Formulas

<p>Let Φ be a uniformly distributed random k-SAT formula with n variables and m clauses. We prove that the Walksat algorithm from Papadimitriou (FOCS 1991)/Schoning (FOCS 1999) finds a satisfying ¨ assignment of Φ in polynomial time w.h.p. if m/n ≤ ρ · 2 k /k for a certain constant ρ > 0. This is an improvement by a factor of Θ(k) over the best previous analysis of Walksat from Coja-Oghlan, Feige, Frieze, Krivelevich, Vilenchik (SODA 2009).</p

Analyzing walksat on random formulas

Abstract. Let Φ be a uniformly distributed random k-SAT formula with n variables and m clauses. We prove that the Walksat algorithm from [16, 17] finds a satisfying assignment of Φ in polynomial time w.h.p. if m/n ≤ ρ · 2 k /k for a certain constant ρ&gt; 0. This is an improvement by a factor of Θ(k) over the best previous analysis of Walksat from [9]. Key words: random structures, phase transitions, k-SAT, local search algorithms.