TY - GEN

T1 - Effectiveness of structural restrictions for hybrid CSPs

AU - Kolmogorov, Vladimir

AU - Rolínek, Michal

AU - Takhanov, Rustem

N1 - Funding Information:
We thank Andrei Krokhin for helpful comments on the manuscript. This work was supported by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 616160.

PY - 2015

Y1 - 2015

N2 - Constraint Satisfaction Problem (CSP) is a fundamental algorithmic problem that appears in many areas of Computer Science. It can be equivalently stated as computing a homomorphism R → Γ between two relational structures, e.g. between two directed graphs. Analyzing its complexity has been a prominent research direction, especially for the fixed template CSPs where the right side Γ is fixed and the left side R is unconstrained. Far fewer results are known for the hybrid setting that restricts both sides simultaneously. It assumes that R belongs to a certain class of relational structures (called a structural restriction in this paper). We study which structural restrictions are effective, i.e. there exists a fixed template Γ (from a certain class of languages) for which the problem is tractable when R is restricted, and NP-hard otherwise. We provide a characterization for structural restrictions that are closed under inverse homomorphisms. The criterion is based on the chromatic number of a relational structure defined in this paper; it generalizes the standard chromatic number of a graph. As our main tool, we use the algebraic machinery developed for fixed template CSPs. To apply it to our case, we introduce a new construction called a “lifted language”. We also give a characterization for structural restrictions corresponding to minor-closed families of graphs, extend results to certain Valued CSPs (namely conservative valued languages), and state implications for (valued) CSPs with ordered variables and for the maximum weight independent set problem on some restricted families of graphs.

AB - Constraint Satisfaction Problem (CSP) is a fundamental algorithmic problem that appears in many areas of Computer Science. It can be equivalently stated as computing a homomorphism R → Γ between two relational structures, e.g. between two directed graphs. Analyzing its complexity has been a prominent research direction, especially for the fixed template CSPs where the right side Γ is fixed and the left side R is unconstrained. Far fewer results are known for the hybrid setting that restricts both sides simultaneously. It assumes that R belongs to a certain class of relational structures (called a structural restriction in this paper). We study which structural restrictions are effective, i.e. there exists a fixed template Γ (from a certain class of languages) for which the problem is tractable when R is restricted, and NP-hard otherwise. We provide a characterization for structural restrictions that are closed under inverse homomorphisms. The criterion is based on the chromatic number of a relational structure defined in this paper; it generalizes the standard chromatic number of a graph. As our main tool, we use the algebraic machinery developed for fixed template CSPs. To apply it to our case, we introduce a new construction called a “lifted language”. We also give a characterization for structural restrictions corresponding to minor-closed families of graphs, extend results to certain Valued CSPs (namely conservative valued languages), and state implications for (valued) CSPs with ordered variables and for the maximum weight independent set problem on some restricted families of graphs.

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U2 - 10.1007/978-3-662-48971-0_48

DO - 10.1007/978-3-662-48971-0_48

M3 - Conference contribution

AN - SCOPUS:84951928007

SN - 9783662489703

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 566

EP - 577

BT - Algorithms and Computation - 26th International Symposium, ISAAC 2015, Proceedings

A2 - Elbassioni, Khaled

A2 - Makino, Kazuhisa

PB - Springer Verlag

T2 - 26th International Symposium on Algorithms and Computation, ISAAC 2015

Y2 - 9 December 2015 through 11 December 2015

ER -