Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials

This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP is not contained in coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT with d literals per clause, to equivalent instances with $O(n^{d-e})$ bits for any e > 0. For the Not-All-Equal SAT problem, a compression to size $\~O(n^{d-1})$ exists. We put these results in a common framework by analyzing the compressibility of binary CSPs. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with $O(n^{2-e})$ bits for any e > 0, unless NP is a subset of coNP/poly.Comment: Updated the cross-composition in lemma 18 (minor update), since the previous version did NOT satisfy requirement 4 of lemma 18 (the proof of Claim 20 was incorrect

Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials

This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP is not contained in coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT with d literals per clause, to equivalent instances with O(n^{d-epsilon}) bits for any epsilon > 0. For the Not-All-Equal SAT problem, a compression to size tilde-O(n^{d-1}) exists. We put these results in a common framework by analyzing the compressibility of binary CSPs. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with O(n^{2-epsilon}) bits for any epsilon > 0, unless NP is contained in coNP/poly

Sketching Cuts in Graphs and Hypergraphs

Sketching and streaming algorithms are in the forefront of current research directions for cut problems in graphs. In the streaming model, we show that $(1-\epsilon)$-approximation for Max-Cut must use $n^{1-O(\epsilon)}$ space; moreover, beating $4/5$-approximation requires polynomial space. For the sketching model, we show that $r$-uniform hypergraphs admit a $(1+\epsilon)$-cut-sparsifier (i.e., a weighted subhypergraph that approximately preserves all the cuts) with $O(\epsilon^{-2} n (r+\log n))$ edges. We also make first steps towards sketching general CSPs (Constraint Satisfaction Problems)

The algebraic structure of the densification and the sparsification tasks for CSPs

The tractability of certain CSPs for dense or sparse instances is known from the 90s. Recently, the densification and the sparsification of CSPs were formulated as computational tasks and the systematical study of their computational complexity was initiated. We approach this problem by introducing the densification operator, i.e. the closure operator that, given an instance of a CSP, outputs all constraints that are satisfied by all of its solutions. According to the Galois theory of closure operators, any such operator is related to a certain implicational system (or, a functional dependency) $\Sigma$. We are specifically interested in those classes of fixed-template CSPs, parameterized by constraint languages $\Gamma$, for which the size of an implicational system $\Sigma$ is a polynomial in the number of variables $n$. We show that in the Boolean case, $\Sigma$ is of polynomial size if and only if $\Gamma$ is of bounded width. For such languages, $\Sigma$ can be computed in log-space or in a logarithmic time with a polynomial number of processors. Given an implicational system $\Sigma$, the densification task is equivalent to the computation of the closure of input constraints. The sparsification task is equivalent to the computation of the minimal key. This leads to ${\mathcal O}({\rm poly}(n)\cdot N^2)$-algorithm for the sparsification task where $N$ is the number of non-redundant sparsifications of an original CSP. Finally, we give a complete classification of constraint languages over the Boolean domain for which the densification problem is tractable

Best-case and worst-case sparsifiability of Boolean CSPs

We continue the investigation of polynomial-time sparsification for NP-complete Boolean Constraint Satisfaction Problems (CSPs). The goal in sparsification is to reduce the number of constraints in a problem instance without changing the answer, such that a bound on the number of resulting constraints can be given in terms of the number of variables n. We investigate how the worst-case sparsification size depends on the types of constraints allowed in the problem formulation (the constraint language). Two algorithmic results are presented. The first result essentially shows that for any arity k, the only constraint type for which no nontrivial sparsification is possible has exactly one falsifying assignment, and corresponds to logical OR (up to negations). Our second result concerns linear sparsification, that is, a reduction to an equivalent instance with O(n) constraints. Using linear algebra over rings of integers modulo prime powers, we give an elegant necessary and sufficient condition for a constraint type to be captured by a degree-1 polynomial over such a ring, which yields linear sparsifications. The combination of these algorithmic results allows us to prove two characterizations that capture the optimal sparsification sizes for a range of Boolean CSPs. For NP-complete Boolean CSPs whose constraints are symmetric (the satisfaction depends only on the number of 1 values in the assignment, not on their positions), we give a complete characterization of which constraint languages allow for a linear sparsification. For Boolean CSPs in which every constraint has arity at most three, we characterize the optimal size of sparsifications in terms of the largest OR that can be expressed by the constraint language

Additive sparsification of CSPs

Multiplicative cut sparsifiers, introduced by Bencz´ur and Karger [STOC’96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA’17] and non-Boolean domains by Butti and Zivn´y [SIDMA’20]. ˇ Bansal, Svensson and Trevisan [FOCS’19] introduced a weaker notion of sparsification termed “additive sparsification”, which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs. As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate P : {0, 1} k → {0, 1} of a fixed arity k, we show that CSP(P) admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP(P) admits an additive sparsifier for any predicate P : Dk → {0, 1} of a fixed arity k on an arbitrary finite domain D

Optimal Polynomial-Time Compression for Boolean Max CSP

In the Boolean maximum constraint satisfaction problem - Max CSP(?) - one is given a collection of weighted applications of constraints from a finite constraint language ?, over a common set of variables, and the goal is to assign Boolean values to the variables so that the total weight of satisfied constraints is maximized. There exists a concise dichotomy theorem providing a criterion on ? for the problem to be polynomial-time solvable and stating that otherwise it becomes NP-hard. We study the NP-hard cases through the lens of kernelization and provide a complete characterization of Max CSP(?) with respect to the optimal compression size. Namely, we prove that Max CSP(?) parameterized by the number of variables n is either polynomial-time solvable, or there exists an integer d ? 2 depending on ?, such that: 1) An instance of Max CSP(?) can be compressed into an equivalent instance with ?(n^d log n) bits in polynomial time, 2) Max CSP(?) does not admit such a compression to ?(n^{d-?}) bits unless NP ? co-NP / poly. Our reductions are based on interpreting constraints as multilinear polynomials combined with the framework of constraint implementations. As another application of our reductions, we reveal tight connections between optimal running times for solving Max CSP(?). More precisely, we show that obtaining a running time of the form ?(2^{(1-?)n}) for particular classes of Max CSPs is as hard as breaching this barrier for Max d-SAT for some d