Using deep learning to construct stochastic local search SAT solvers with performance bounds

The Boolean Satisfiability problem (SAT) is the most prototypical NP-complete problem and of great practical relevance. One important class of solvers for this problem are stochastic local search (SLS) algorithms that iteratively and randomly update a candidate assignment. Recent breakthrough results in theoretical computer science have established sufficient conditions under which SLS solvers are guaranteed to efficiently solve a SAT instance, provided they have access to suitable "oracles" that provide samples from an instance-specific distribution, exploiting an instance's local structure. Motivated by these results and the well established ability of neural networks to learn common structure in large datasets, in this work, we train oracles using Graph Neural Networks and evaluate them on two SLS solvers on random SAT instances of varying difficulty. We find that access to GNN-based oracles significantly boosts the performance of both solvers, allowing them, on average, to solve 17% more difficult instances (as measured by the ratio between clauses and variables), and to do so in 35% fewer steps, with improvements in the median number of steps of up to a factor of 8. As such, this work bridges formal results from theoretical computer science and practically motivated research on deep learning for constraint satisfaction problems and establishes the promise of purpose-trained SAT solvers with performance guarantees.Comment: 15 pages, 9 figures, code available at https://github.com/porscheofficial/sls_sat_solving_with_deep_learnin

Proof Complexity Meets Algebra

We analyse how the standard reductions between constraint satisfaction problems affect their proof complexity. We show that, for the most studied propositional and semi-algebraic proof systems, the classical constructions of pp-interpretability, homomorphic equivalence and addition of constants to a core preserve the proof complexity of the CSP. As a result, for those proof systems, the classes of constraint languages for which small unsatisfiability certificates exist can be characterised algebraically. We illustrate our results by a gap theorem saying that a constraint language either has resolution refutations of bounded width, or does not have bounded-depth Frege refutations of subexponential size. The former holds exactly for the widely studied class of constraint languages of bounded width. This class is also known to coincide with the class of languages with Sums-of-Squares refutations of sublinear degree, a fact for which we provide an alternative proof. We hence ask for the existence of a natural proof system with good behaviour with respect to reductions and simultaneously small size refutations beyond bounded width. We give an example of such a proof system by showing that bounded-degree Lovasz-Schrijver satisfies both requirements

Approximation Limits of Linear Programs (Beyond Hierarchies)

We develop a framework for approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n^{1/2-eps})-approximations for CLIQUE require linear programs of size 2^{n^\Omega(eps)}. (This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem.) Moreover, we establish a similar result for approximations of semidefinite programs by linear programs. Our main ingredient is a quantitative improvement of Razborov's rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of certain perturbations of the unique disjointness matrix.Comment: 23 pages, 2 figure

Hardness of submodular cost allocation : lattice matching and a simplex coloring conjecture

We consider the Minimum Submodular Cost Allocation (MSCA) problem. In this problem, we are given k submodular cost functions f1, ... , fk: 2V -> R+ and the goal is to partition V into k sets A1, ..., Ak so as to minimize the total cost sumi = 1,k fi(Ai). We show that MSCA is inapproximable within any multiplicative factor even in very restricted settings; prior to our work, only Set Cover hardness was known. In light of this negative result, we turn our attention to special cases of the problem. We consider the setting in which each function fi satisfies fi = gi + h, where each gi is monotone submodular and h is (possibly non-monotone) submodular. We give an O(k log |V|) approximation for this problem. We provide some evidence that a factor of k may be necessary, even in the special case of HyperLabel. In particular, we formulate a simplex-coloring conjecture that implies a Unique-Games-hardness of (k - 1 - epsilon) for k-uniform HyperLabel and label set [k]. We provide a proof of the simplex-coloring conjecture for k=3

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