TY - JOUR

T1 - Best-case and worst-case sparsifiability of Boolean CSPs

AU - Chen, Hubie

AU - Jansen, Bart M.P.

AU - Pieterse, Astrid

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Computational Complexity

KW - Data Structures and Algorithms

U2 - 10.48550/arXiv.1809.06171

DO - 10.48550/arXiv.1809.06171

M3 - Article

VL - 2018

JO - arXiv

JF - arXiv

M1 - 1809.06171v1

ER -