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 \({\mathbf {R}\rightarrow \varvec{\Gamma }}\) 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 \(\varvec{\Gamma }\) is fixed and the left side \(\mathbf {R}\) is unconstrained.
Far fewer results are known for the hybrid setting that restricts both sides simultaneously. It assumes that \(\mathbf {R}\) 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 \(\varvec{\Gamma }\) (from a certain class of languages) for which the problem is tractable when \(\mathbf {R}\) 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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Notes
- 1.
We will allow two possibilities: (i) weights are positive integers, and the length of the description of \(\mathcal{I}\) grows linearly with \(w(f,\mathbf {v})\); (ii) weights are positive rationals. All our statements for VCSPs will hold under both models. Note that in the literature weights \(w(f,\mathbf {v})\) are usually omitted, and T is allowed to be a multiset rather than a set; this is equivalent to model (i). Including weights will be convenient for hybrid VCSPs.
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Acknowledgements
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.
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Kolmogorov, V., Rolínek, M., Takhanov, R. (2015). Effectiveness of Structural Restrictions for Hybrid CSPs. In: Elbassioni, K., Makino, K. (eds) Algorithms and Computation. ISAAC 2015. Lecture Notes in Computer Science(), vol 9472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48971-0_48
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