Abstract

We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable-constraint incidence graph of the CSP instance. We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k, and we attempt to fully classify them into the following three cases: 1. The exact optimum can be computed in FPT time. 2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPTAS), which computes a $(1-\epsilon)$-approximation in time $f(k,\epsilon)\cdot poly(n)$. 3. There is no FPTAS unless FPT=W[1]. For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results.

Highlights

  • Constraint Satisfaction Problems (CSPs) play a central role in almost all branches of theoretical computer science

  • For the most part, the standard parameterized complexity script: we consider the input instance’s incidence graph and try to determine the complexity of the CSP when parameterized by various graph

  • The new ingredient in our approach is that, in addition to trying to determine which parameters make a CSP fixed-parameter tractable (FPT) or W-hard, we ask if the optimization versions of W-hard cases can be well-approximated

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Summary

Introduction

Constraint Satisfaction Problems (CSPs) play a central role in almost all branches of theoretical computer science. Starting from CNF-SAT, the prototypical NP-complete problem, the computational complexity of CSPs has been widely studied from various points of view. Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany. In this paper we focus on two aspects of CSP complexity which, though extremely wellinvestigated, have mostly been considered separately so far in the literature: parameterized complexity and approximability.

Parameterized CSPs
Approximation
Results
See the next section for a definition of incidence graphs
Preliminaries
CNF-SAT and MAX-CNF-SAT
Approximation Algorithm parameterized by clique-width
Hardness parameterized by neighborhood diversity
From Treewidth to Clique-width
Majority and Threshold CSPs
Hardness of exact algorithms
Exact Algorithm parameterized by vertex cover
Approximation Algorithm parameterized by feedback vertex set
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