### Abstract

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.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 566-577 |

Number of pages | 12 |

Volume | 9472 |

ISBN (Print) | 9783662489703 |

DOIs | |

Publication status | Published - 2015 |

Event | 26th International Symposium on Algorithms and Computation, ISAAC 2015 - Nagoya, Japan Duration: Dec 9 2015 → Dec 11 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9472 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 26th International Symposium on Algorithms and Computation, ISAAC 2015 |
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Country | Japan |

City | Nagoya |

Period | 12/9/15 → 12/11/15 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 9472, pp. 566-577). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9472). Springer Verlag. https://doi.org/10.1007/978-3-662-48971-0_48

**Effectiveness of structural restrictions for hybrid CSPs.** / Kolmogorov, Vladimir; Rolínek, Michal; Takhanov, Rustem.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 9472, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9472, Springer Verlag, pp. 566-577, 26th International Symposium on Algorithms and Computation, ISAAC 2015, Nagoya, Japan, 12/9/15. https://doi.org/10.1007/978-3-662-48971-0_48

}

TY - GEN

T1 - Effectiveness of structural restrictions for hybrid CSPs

AU - Kolmogorov, Vladimir

AU - Rolínek, Michal

AU - Takhanov, Rustem

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.

UR - http://www.scopus.com/inward/record.url?scp=84951928007&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84951928007&partnerID=8YFLogxK

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

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

M3 - Conference contribution

SN - 9783662489703

VL - 9472

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

SP - 566

EP - 577

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

PB - Springer Verlag

ER -