TY - GEN

T1 - Optimal polynomial-time compression for boolean max CSP

AU - Jansen, Bart M.P.

AU - Wlodarczyk, Michal

PY - 2020/8/1

Y1 - 2020/8/1

N2 - In the Boolean maximum constraint satisfaction problem – Max CSP(Γ) – one is given a collection of weighted applications of constraints from a finite constraint language Γ, 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 Γ 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(Γ) with respect to the optimal compression size. Namely, we prove that Max CSP(Γ) parameterized by the number of variables n is either polynomial-time solvable, or there exists an integer d ≥ 2 depending on Γ, such that: 1. An instance of Max CSP(Γ) can be compressed into an equivalent instance with O(nd log n) bits in polynomial time, 2. Max CSP(Γ) does not admit such a compression to O(nd−ε) 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(Γ). More precisely, we show that obtaining a running time of the form O(2(1−ε)n) for particular classes of Max CSPs is as hard as breaching this barrier for Max d-SAT for some d.

AB - In the Boolean maximum constraint satisfaction problem – Max CSP(Γ) – one is given a collection of weighted applications of constraints from a finite constraint language Γ, 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 Γ 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(Γ) with respect to the optimal compression size. Namely, we prove that Max CSP(Γ) parameterized by the number of variables n is either polynomial-time solvable, or there exists an integer d ≥ 2 depending on Γ, such that: 1. An instance of Max CSP(Γ) can be compressed into an equivalent instance with O(nd log n) bits in polynomial time, 2. Max CSP(Γ) does not admit such a compression to O(nd−ε) 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(Γ). More precisely, we show that obtaining a running time of the form O(2(1−ε)n) for particular classes of Max CSPs is as hard as breaching this barrier for Max d-SAT for some d.

KW - constraint satisfaction problem

KW - Exponential time algorithms

KW - Kernelization

UR - http://www.scopus.com/inward/record.url?scp=85092527873&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2020.63

DO - 10.4230/LIPIcs.ESA.2020.63

M3 - Conference contribution

AN - SCOPUS:85092527873

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 28th Annual European Symposium on Algorithms, ESA 2020

A2 - Grandoni, Fabrizio

A2 - Herman, Grzegorz

A2 - Sanders, Peter

PB - Schloss Dagstuhl - Leibniz-Zentrum für Informatik

T2 - 28th Annual European Symposium on Algorithms, ESA 2020

Y2 - 7 September 2020 through 9 September 2020

ER -