The Constraint Satisfaction Problem: Complexity and Approximability
DFU - Vol. 7
Dagstuhl Follow-Ups
DFU
https://www.dagstuhl.de/dagpub/1868-8977
https://dblp.org/db/series/dfu
1868-8977
Andrei
Krokhin
Andrei Krokhin
Stanislav
Zivny
Stanislav Zivny
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
7
2017
978-3-95977-003-3
https://www.dagstuhl.de/dagpub/978-3-95977-003-3
Front Matter, Table of Contents, Preface, List of Authors
Front Matter, Table of Contents, Preface, List of Authors
Front Matter
Table of Contents
Preface
List of Authors
0:i-0:xii
Regular Paper
Andrei
Krokhin
Andrei Krokhin
Stanislav
Zivny
Stanislav Zivny
10.4230/DFU.Vol7.15301.i
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Polymorphisms, and How to Use Them
This article describes the algebraic approach to Constraint Satisfaction Problem that led to many developments in both CSP and universal algebra. No prior knowledge of universal algebra is assumed.
Constraint satisfaction
Complexity
Universal algebra
Polymorphism
1-44
Regular Paper
Libor
Barto
Libor Barto
Andrei
Krokhin
Andrei Krokhin
Ross
Willard
Ross Willard
10.4230/DFU.Vol7.15301.1
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Absorption in Universal Algebra and CSP
The algebraic approach to Constraint Satisfaction Problem led to many developments in both CSP and universal algebra. The notion of absorption was successfully applied on both sides of the connection. This article introduces the concept of absorption, illustrates its use in a number of basic proofs and provides an overview of the most important results obtained by using it.
Constraint satisfaction problem
Algebraic approach
Absorption
45-77
Regular Paper
Libor
Barto
Libor Barto
Marcin
Kozik
Marcin Kozik
10.4230/DFU.Vol7.15301.45
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Constraint Satisfaction Problems over Numeric Domains
We present a survey of complexity results for constraint satisfaction problems (CSPs) over the integers, the rationals, the reals, and the complex numbers. Examples of such problems are feasibility of linear programs, integer linear programming, the max-atoms problem, Hilbert's tenth problem, and many more. Our particular focus is to identify those CSPs that can be solved in polynomial time, and to distinguish them from CSPs that are NP-hard. A very helpful tool for obtaining complexity classifications in this context is the concept of a polymorphism from universal algebra.
Constraint satisfaction problems
Numerical domains
79-111
Regular Paper
Manuel
Bodirsky
Manuel Bodirsky
Marcello
Mamino
Marcello Mamino
10.4230/DFU.Vol7.15301.79
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Hybrid Tractable Classes of Constraint Problems
We present a survey of complexity results for hybrid constraint satisfaction problems (CSPs) and valued constraint satisfaction problems (VCSPs). These are classes of (V)CSPs defined by restrictions that are not exclusively language-based or structure-based.
Constraint satisfaction problems
Optimisation
Tractability
113-135
Regular Paper
Martin C.
Cooper
Martin C. Cooper
Stanislav
Zivny
Stanislav Zivny
10.4230/DFU.Vol7.15301.113
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Backdoor Sets for CSP
A backdoor set of a CSP instance is a set of variables whose instantiation moves the instance into a fixed class of tractable instances (an island of tractability). An interesting algorithmic task is to find a small backdoor set efficiently: once it is found we can solve the instance by solving a number of tractable instances. Parameterized complexity provides an adequate framework for studying and solving this algorithmic task, where the size of the backdoor set provides a natural parameter. In this survey we present some recent parameterized complexity results on CSP backdoor sets, focusing on backdoor sets into islands of tractability that are defined in terms of constraint languages.
Backdoor sets
Constraint satisfaction problems
Parameterized complexity
Polymorphisms
137-157
Regular Paper
Serge
Gaspers
Serge Gaspers
Sebastian
Ordyniak
Sebastian Ordyniak
Stefan
Szeider
Stefan Szeider
10.4230/DFU.Vol7.15301.137
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
On the Complexity of Holant Problems
In this article we survey recent developments on the complexity of Holant problems. We discuss three different aspects of Holant problems: the decision version, exact counting, and approximate counting.
Computational complexity
Counting complexity
Dichotomy theorems
Approximate counting
Holant problems
159-177
Regular Paper
Heng
Guo
Heng Guo
Pinyan
Lu
Pinyan Lu
10.4230/DFU.Vol7.15301.159
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Parameterized Constraint Satisfaction Problems: a Survey
We consider constraint satisfaction problems parameterized above or below guaranteed values. One example is MaxSat parameterized above m/2: given a CNF formula F with m clauses, decide whether there is a truth assignment that satisfies at least m/2 + k clauses, where k is the parameter. Among other problems we deal with are MaxLin2-AA (given a system of linear equations over F_2 in which each equation has a positive integral weight, decide whether there is an assignment to the variables that satisfies equations of total weight at least W/2+k, where W is the total weight of all equations), Max-r-Lin2-AA (the same as MaxLin2-AA, but each equation has at most r variables, where r is a constant) and Max-r-Sat-AA (given a CNF formula F with m clauses in which each clause has at most r literals, decide whether there is a truth assignment satisfying at least sum_{i=1}^m (1-2^{r_i})+k clauses, where k is the parameter, r_i is the number of literals in clause i, and r is a constant). We also consider Max-r-CSP-AA, a natural generalization of both Max-r-Lin2-AA and Max-r-Sat-AA, order (or, permutation) constraint satisfaction problems parameterized above the average value and some other problems related to MaxSat. We discuss results, both polynomial kernels and parameterized algorithms, obtained for the problems mainly in the last few years as well as some open questions.
Constraint satisfaction problems
Fixed-parameter tractability
179-203
Regular Paper
Gregory
Gutin
Gregory Gutin
Anders
Yeo
Anders Yeo
10.4230/DFU.Vol7.15301.179
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Counting Constraint Satisfaction Problems
This chapter surveys counting Constraint Satisfaction Problems (counting CSPs, or #CSPs) and their computational complexity. It aims to provide an introduction to the main concepts and techniques, and present a representative selection of results and open problems. It does not cover holants, which are the subject of a separate chapter.
Approximation algorithms
Computational complexity
Constraint satisfaction problems
Counting problems
Partition functions
205-231
Regular Paper
Mark
Jerrum
Mark Jerrum
10.4230/DFU.Vol7.15301.205
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
The Complexity of Valued CSPs
We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems (VCSPs). We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify general features of the complexity landscape.
Constraint satisfaction problems
Optimisation
Tractability
233-266
Regular Paper
Andrei
Krokhin
Andrei Krokhin
Stanislav
Zivny
Stanislav Zivny
10.4230/DFU.Vol7.15301.233
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Algebra and the Complexity of Digraph CSPs: a Survey
We present a brief survey of some of the key results on the interplay between algebraic and graph-theoretic methods in the study of the complexity of digraph-based constraint satisfaction problems.
Constraint satisfaction problems
Polymorphisms
Digraphs
267-285
Regular Paper
Benoit
Larose
Benoit Larose
10.4230/DFU.Vol7.15301.267
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Approximation Algorithms for CSPs
In this survey, we offer an overview of approximation algorithms for constraint satisfaction problems (CSPs) - we describe main results and discuss various techniques used for solving CSPs.
Constraint satisfaction problems
Approximation algorithms
SDP
UGC
287-325
Regular Paper
Konstantin
Makarychev
Konstantin Makarychev
Yury
Makarychev
Yury Makarychev
10.4230/DFU.Vol7.15301.287
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Quantified Constraints in Twenty Seventeen
I present a survey of recent advances in the algorithmic and computational complexity theory of non-Boolean Quantified Constraint Satisfaction Problems, incorporating some more modern research directions.
Quantified constraints
Constraint satisfaction problems
Computational complexity
Parameterized complexity
Universal algebra
327-346
Regular Paper
Barnaby
Martin
Barnaby Martin
10.4230/DFU.Vol7.15301.327
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode