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DOI: 10.4230/LIPIcs.ISAAC.2017.65

Hybrid VCSPs with Crisp and Valued Conservative Templates

Authors: Rustem Takhanov

Published in: LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

Abstract

A constraint satisfaction problem (CSP) is a problem of computing a homomorphism R -> G between two relational structures, e.g. between two directed graphs. Analyzing its complexity has been a very fruitful research direction, especially for fixed template CSPs (or, non-uniform CSPs), denoted CSP(G), in which the right side structure G is fixed and the left side structure R is unconstrained. Recently, the hybrid setting, written CSP_H(G), where both sides are restricted simultaneously, attracted some attention. It assumes that R is taken from a class of relational structures H (called the structural restriction) that additionally is closed under inverse homomorphisms. The last property allows to exploit an algebraic machinery that has been developed for fixed template CSPs. The key concept that connects hybrid CSPs with fixed-template CSPs is the so called lifted language. Namely, this is a constraint language G_R that can be constructed from an input R. The tractability of the language G_R for any input R from H is a necessary condition for the tractability of the hybrid problem. In the first part we investigate templates G for which the latter condition is not only necessary, but also is sufficient. We call such templates G widely tractable. For this purpose, we construct from G a new finite relational structure G' and define a maximal structural restriction H_0 as a class of structures homomorphic to G'. For the so called strongly BJK templates that probably captures all templates, we prove that wide tractability is equivalent to the tractability of CSP_{H_0}(G). Our proof is based on the key observation that R is homomorphic to G' if and only if the core of G_R is preserved by a Siggers polymorphism. Analogous result is shown for conservative valued CSPs.

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Rustem Takhanov. Hybrid VCSPs with Crisp and Valued Conservative Templates. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 65:1-65:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{takhanov:LIPIcs.ISAAC.2017.65,
  author =	{Takhanov, Rustem},
  title =	{{Hybrid VCSPs with Crisp and Valued Conservative Templates}},
  booktitle =	{28th International Symposium on Algorithms and Computation (ISAAC 2017)},
  pages =	{65:1--65:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-054-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{92},
  editor =	{Okamoto, Yoshio and Tokuyama, Takeshi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.65},
  URN =		{urn:nbn:de:0030-drops-82474},
  doi =		{10.4230/LIPIcs.ISAAC.2017.65},
  annote =	{Keywords: constraint satisfaction problem, polymorphisms, algebraic approach, lifted language, hybrid CSPs, closed under inverse homomorphisms}
}

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DOI: 10.4230/LIPIcs.STACS.2010.2493

A Dichotomy Theorem for the General Minimum Cost Homomorphism Problem

Authors: Rustem Takhanov

Published in: LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Abstract

In the constraint satisfaction problem ($CSP$), the aim is to find an assignment of values to a set of variables subject to specified constraints. In the minimum cost homomorphism problem ($MinHom$), one is additionally given weights $c_{va}$ for every variable $v$ and value $a$, and the aim is to find an assignment $f$ to the variables that minimizes $\sum_{v} c_{vf(v)}$. Let $MinHom\left( \Gamma \right)$ denote the $MinHom$ problem parameterized by the set of predicates allowed for constraints. $MinHom\left( \Gamma \right)$ is related to many well-studied combinatorial optimization problems, and concrete applications can be found in, for instance, defence logistics and machine learning. We show that $MinHom\left( \Gamma \right)$ can be studied by using algebraic methods similar to those used for CSPs. With the aid of algebraic techniques, we classify the computational complexity of $MinHom\left( \Gamma \right)$ for all choices of $\Gamma$. Our result settles a general dichotomy conjecture previously resolved only for certain classes of directed graphs, [Gutin, Hell, Rafiey, Yeo, European J. of Combinatorics, 2008].

Cite as

Rustem Takhanov. A Dichotomy Theorem for the General Minimum Cost Homomorphism Problem. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 657-668, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{takhanov:LIPIcs.STACS.2010.2493,
  author =	{Takhanov, Rustem},
  title =	{{A Dichotomy Theorem for the General Minimum Cost Homomorphism Problem}},
  booktitle =	{27th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{657--668},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-16-3},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{5},
  editor =	{Marion, Jean-Yves and Schwentick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2493},
  URN =		{urn:nbn:de:0030-drops-24936},
  doi =		{10.4230/LIPIcs.STACS.2010.2493},
  annote =	{Keywords: Minimum cost homomorphisms problem, relational clones, constraint satisfaction problem, perfect graphs, supervised learning}
}

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Hybrid VCSPs with Crisp and Valued Conservative Templates

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A Dichotomy Theorem for the General Minimum Cost Homomorphism Problem

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