Boolean CSPs Research Articles

Sparsification of Binary CSPs

A cut $\varepsilon$-sparsifier of a weighted graph $G$ is a re-weighted subgraph of $G$ of (quasi)linear size that preserves the size of all cuts up to a multiplicative factor of $\varepsilon$. Since their introduction by Bencz\'ur and Karger [STOC'96], cut sparsifiers have proved extremely influential and found various applications. Going beyond cut sparsifiers, Filtser and Krauthgamer [SIDMA'17] gave a precise classification of which binary Boolean CSPs are sparsifiable. In this paper, we extend their result to binary CSPs on arbitrary finite domains.

Even Delta-Matroids and the Complexity of Planar Boolean CSPs

The main result of this article is a generalization of the classical blossom algorithm for finding perfect matchings. Our algorithm can efficiently solve Boolean CSPs where each variable appears in exactly two constraints (we call it edge CSP) and all constraints are even Δ-matroid relations (represented by lists of tuples). As a consequence of this, we settle the complexity classification of planar Boolean CSPs started by Dvořák and Kupec. Using a reduction to even Δ-matroids, we then extend the tractability result to larger classes of Δ-matroids that we call efficiently coverable . It properly includes classes that were known to be tractable before, namely, co-independent , compact , local , linear , and binary , with the following caveat: We represent Δ-matroids by lists of tuples, while the last two use a representation by matrices. Since an n × n matrix can represent exponentially many tuples, our tractability result is not strictly stronger than the known algorithm for linear and binary Δ-matroids.

LP/SDP Hierarchy Lower Bounds for Decoding Random LDPC Codes

Random $(d_{v},d_{c})$ - regular low-density parity-check (LDPC) codes, where each variable is involved in $d_{v}$ parity checks and each parity check involves $d_{c}$ variables are well-known to achieve the Shannon capacity of the binary symmetric channel, for sufficiently large $d_{v}$ and $d_{c}$ , under exponential time decoding. However, polynomial time algorithms are only known to correct a much smaller fraction of errors. One of the most powerful polynomial-time algorithms with a formal analysis is the linear programming (LP) decoding algorithm of Feldman et al., which is known to correct an $\Omega (1/d_{c})$ fraction of errors. In this paper, we show that fairly powerful extensions of LP decoding, based on the Sherali–Adams and Lasserre hierarchies, fail to correct much more errors than the basic LP-decoder. In particular, we show that: 1) for any values of $d_{v}$ and $d_{c}$ , a linear number of rounds of the Sherali–Adams LP hierarchy cannot correct more than an $O(1/d_{c})$ fraction of errors on a random $(d_{v},d_{c})$ -regular LDPC code; and 2) for any value of $d_{v}$ and infinitely many values of $d_{c}$ , a linear number of rounds of the Lasserre SDP hierarchy cannot correct more than an $O(1/d_{c})$ fraction of errors on a random $(d_{v},d_{c})$ -regular LDPC code. Our proofs use a new stretching and collapsing technique that allows us to leverage recent progress in the study of the limitations of LP/SDP hierarchies for Maximum Constraint Satisfaction Problems (Max-CSPs). The problem then reduces to the construction of special balanced pairwise independent distributions for Sherali–Adams and special cosets of balanced pairwise independent subgroups for Lasserre. Our (algebraic) construction for the Lasserre hierarchy is based on designing sets of points in ${\mathbb F}_{q}^{d}$ (for $q$ any power of 2 and $d = 2,3$ ) with special hyperplane-incidence properties—constructions that may be of independent interest. An intriguing consequence of our work is that expansion seems to be both the strength and the weakness of random regular LDPC codes. Some of our techniques are more generally applicable to a large class of Boolean CSPs called Min-Ones. In particular, for $k$ -Hypergraph Vertex Cover, we obtain an improved integrality gap of $k-1-\epsilon $ that holds after a linear number of rounds of the Lasserre hierarchy, for any $k = q+1$ with $q$ an arbitrary prime power. The best previous gap for a linear number of rounds was equal to $2-\epsilon $ and due to Schoenebeck.

Gap theorems for robust satisfiability: Boolean CSPs and beyond

A computational problem exhibits a “gap property” when there is no tractable boundary between two disjoint sets of instances. We establish a Gap Trichotomy Theorem for a family of constraint problem variants, completely classifying the complexity of possible NP-hard gaps in the case of Boolean domains. As a consequence, we obtain a number of dichotomies for the complexity of specific variants of the constraint satisfaction problem: all are either polynomial-time tractable or NP-complete. Schaefer's original dichotomy for SAT variants is a notable particular case.Universal algebraic methods have been central to recent efforts in classifying the complexity of constraint satisfaction problems. A second contribution of the article is to develop aspects of the algebraic approach in the context of a number of variants of the constraint satisfaction problem. In particular, this allows us to lift our results on Boolean domains to many templates on non-Boolean domains.

PCPs via the low-degree long code and hardness for constrained hypergraph coloring

We develop new techniques to incorporate the recently proposed “short code” (a low-degree version of the long code) into the construction and analysis of PCPs in the classical “label-cover + Fourier Analysis” framework. As a result, we obtain more size-efficient PCPs that yield improved hardness results for approximating CSPs and certain coloring-type problems. In particular, we show a hardness for a variant of hypergraph coloring (with hyperedges of size 6), with a gap between 2 and $$\exp ({2^{\Omega (\sqrt {\log \log N} )}})$$ number of colors where N is the number of vertices. This is the first hardness result to go beyond the O(logN) barrier for a coloring-type problem. Our hardness bound is a doubly exponential improvement over the previously known O(log logN)-coloring hardness for 2-colorable hypergraphs, and an exponential improvement over the $${(\log N)^{\Omega (1)}}$$ -coloring hardness for O(1)-colorable hypergraphs. Stated in terms of “covering complexity,” we show that for 6-ary Boolean CSPs, it is hard to decide if a given instance is perfectly satisfiable or if it requires more than $${2^{\Omega (\sqrt {\log \log N} )}}$$ assignments for covering all of the constraints. While our methods do not yield a result for conventional hypergraph coloring due to some technical reasons, we also prove hardness of $${(\log N)^{\Omega (1)}}$$ -coloring 2-colorable 8-uniform hypergraphs (this result relies just on the long code). A key algebraic result driving our analysis concerns a very low-soundness error testing method for Reed-Muller codes. We prove that if a function $$\beta :{\text{F}}_2^m \to {{\text{F}}_2}$$ is $${2^{\Omega (d)}}$$ far in absolute distance from polynomials of degree m − d, then the probability that deg(β g ) ≤ m − 3d/4 for a random degree d/4 polynomial g is doubly exponentially small in d.

Approximate counting for complex-weighted Boolean constraint satisfaction problems

Constraint satisfaction problems (or CSPs) have been extensively studied in, for instance, artificial intelligence, database theory, graph theory, and statistical physics. From a practical viewpoint, it is beneficial to approximately solve those CSPs. When one tries to approximate the total number of truth assignments that satisfy all Boolean-valued constraints for (unweighted) Boolean CSPs, there is a known trichotomy theorem by which all such counting problems are neatly classified into exactly three categories under polynomial-time (randomized) approximation-preserving reductions. In contrast, we obtain a dichotomy theorem of approximate counting for complex-weighted Boolean CSPs, provided that all complex-valued unary constraints are freely available to use. It is the expressive power of free unary constraints that enables us to prove such a stronger, complete classification theorem. This discovery makes a step forward in the quest for the approximation-complexity classification of all counting CSPs. To deal with complex weights, we employ proof techniques of factorization and arity reduction along the line of solving Holant problems. Moreover, we introduce a novel notion of T-constructibility that naturally induces approximation-preserving reducibility. Our result also gives an approximation analogue of the dichotomy theorem on the complexity of exact counting for complex-weighted Boolean CSPs.

A dichotomy theorem for the approximate counting of complex-weighted bounded-degree Boolean CSPs

We determine the computational complexity of approximately counting the total weight of variable assignments for every complex-weighted Boolean constraint satisfaction problem (or CSP) with any number of additional unary (i.e., arity 1) constraints, particularly, when degrees of input instances are bounded from above by a fixed constant. All degree-1 counting CSPs are obviously solvable in polynomial time. When the instance’s degree is more than two, we present a dichotomy theorem that classifies all counting CSPs admitting free unary constraints into exactly two categories. This classification theorem extends, to complex-weighted problems, an earlier result on the approximation complexity of unweighted counting Boolean CSPs of bounded degree. The framework of the proof of our theorem is based on a theory of signature developed from Valiant’s holographic algorithms that can efficiently solve seemingly intractable counting CSPs. Despite the use of arbitrary complex weight, our proof of the classification theorem is rather elementary and intuitive due to an extensive use of a novel notion of limited T-constructibility. For the remaining degree-2 problems, in contrast, they are as hard to approximate as Holant problems, which are a generalization of counting CSPs.

The SAT–UNSAT transition for random constraint satisfaction problems

We study threshold phenomena for a large class of random constraint satisfaction problems over finite domains. Our main contribution is a complete classification of the nature (sharp or coarse) of the SAT–UNSAT transition for random Boolean CSPs, which is based on easily decidable properties.

Expected number of locally maximal solutions for random Boolean CSPs

For a large number of random Boolean constraint satisfaction problems, such as random $k$-SAT, we study how the number of locally maximal solutions evolves when constraints are added. We give the exponential order of the expected number of these distinguished solutions and prove it depends on the sensitivity of the allowed constraint functions only. As a by-product we provide a general tool for computing an upper bound of the satisfiability threshold for any problem of a large class of random Boolean CSPs.

Combinatorial sharpness criterion and phase transition classification for random CSPs

We investigate the nature of the phase transition (sharp or coarse) for random constraint satisfaction problems. We first give a sharp threshold criterion specified for CSPs, which is derived from Friedgut–Bourgain’s one. Thus, we get a complete and precise classification of the nature of the threshold for symmetric Boolean CSPs. In particular we show that it is governed by two local properties strongly related to the problems 1−SAT and 2−XOR−SAT .