The CSP Dichotomy Holds for Digraphs with No Sources and No Sinks (A Positive Answer to a Conjecture of Bang-Jensen and Hell)

Barto, Libor; Kozik, Marcin; Niven, Todd

doi:10.1137/070708093

Cited by 136 publications

(166 citation statements)

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“…Barto, Kozik and Niven [7] generalized Hell and Nešetřil's classification to directed graphs with no sources or sinks. For so-called maximal constraint languages and conservative constraint languages classifications were obtained in [18] and [14] (see also [4]) respectively.…”

Section: Conjecture 33 (The Dichotomy Conjecture) Let γ Be a Finitementioning

confidence: 99%

On Some Combinatorial Optimization Problems : Algorithms and Complexity

Uppman¹

2015

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This thesis is about the computational complexity of several classes of combinatorial optimization problems, all related to the constraint satisfaction problems.A constraint language consists of a domain and a set of relations on the domain. For each such language there is a constraint satisfaction problem (CSP). In this problem we are given a set of variables and a collection of constraints, each of which is constraining some variables with a relation in the language. The goal is to determine if domain values can be assigned to the variables in a way that satisfies all constraints. An important question is for which constraint languages the corresponding CSP can be solved in polynomial time. We study this kind of question for optimization problems related to the CSPs.The main focus is on extended minimum cost homomorphism problems. These are optimization versions of CSPs where instances come with an objective function given by a weighted sum of unary cost functions, and where the goal is not only to determine if a solution exists, but to find one of minimum cost. We prove a complete classification of the complexity for these problems on three-element domains. We also obtain a classification for the so-called conservative case.Another class of combinatorial optimization problems are the surjective maximum CSPs. These problems are variants of CSPs where a non-negative weight is attached to each constraint, and the objective is to find a surjective mapping of the variables to values that maximizes the weighted sum of satisfied constraints. The surjectivity requirement causes these problems to behave quite different from for example the minimum cost homomorphism problems, and many powerful techniques are not applicable. We prove a dichotomy for the complexity of the problems in this class on two-element domains. An essential ingredient in the proof is an algorithm that solves a generalized version of the minimum cut problem. This algorithm might be of independent interest.In a final part we study properties of NP-hard optimization problems. This is done with the aid of restricted forms of polynomial-time reductions that for example preserves solvability in sub-exponential time. Two classes of iv optimization problems similar to those discussed above are considered, and for both we obtain what may be called an easiest NP-hard problem. We also establish some connections to the exponential time hypothesis.This work has been supported in part by the National Graduate School in Computer Science (CUGS), Sweden. Populärvetenskaplig sammanfattningOptimering går ut på att hitta värden för ett antal variabler så att resultatet blir så bra som möjligt (enligt en given kostnadsfunktion). Vi studerar kombinatoriska optimeringsproblem, problem där man för varje variabel enbart har ett ändligt antal möjliga val. Sådana problem är i någon mening lätta; för att hitta en så bra lösning som möjligt kan man helt enkelt prova alla sätt att välja värden för variablerna. Tyvärr är det här inte någon metod som fungerar i praktiken...

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Section: Conjecture 33 (The Dichotomy Conjecture) Let γ Be a Finitementioning

confidence: 99%

On Some Combinatorial Optimization Problems : Algorithms and Complexity

Uppman¹

2015

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“…This dichotomy conjecture remains unsettled, although dichotomy is now known on substantial classes (e.g. structures of size ≤ 3 [19,6] and smooth digraphs [12,2]). Various methods, combinatorial (graph-theoretic), logical and universal-algebraic have been brought to bear on this classification project, with many remarkable consequences.…”

Section: Introductionmentioning

confidence: 99%

“…For example, it is known precisely which smooth digraphs have a weak near unanimity polymorphism [2] and which digraphs enjoy Mal'cev [8] This paper is organised as follows. After the preliminaries we deal with upper bounds and essential polymorphisms in Section 3.…”

Section: Introductionmentioning

confidence: 99%

QCSP on Semicomplete Digraphs

Đapić

Marković

Martin

2014

Automata, Languages, and Programming

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Abstract. We study the (non-uniform) quantified constraint satisfaction problem QCSP(H) as H ranges over semicomplete digraphs. We obtain a complexity-theoretic trichotomy: QCSP(H) is either in P, is NP-complete or is Pspace-complete. The largest part of our work is the algebraic classification of precisely which semicompletes enjoy only essentially unary polymorphisms, which is combinatorially interesting in its own right.

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“…It is mentioned there that combining the results of [2] and [5], and using theorems from [1], one can show that if a finite algebra generates a congruence join semidistributive variety, then it has cyclic terms. In this paper we present a new and direct proof of this fact.…”

Section: Introductionmentioning

confidence: 99%