We investigate the List H-Coloring problem, the generalization of graph coloring that asks whether an input graph G admits a homomorphism to the undirected graph H (possibly with loops), such that each vertex v ∈ V(G) is mapped to a vertex on its list L(v) ⊆ V(H). An important result by Feder, Hell, and Huang [JGT 2003] states that List H-Coloring is polynomial-time solvable if H is a so-called bi-arc graph, and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an n-vertex instance be efficiently reduced to an equivalent instance of bitsize 𝒪(n^(2-ε)) for some ε > 0? We prove that if H is not a bi-arc graph, then List H-Coloring does not admit such a sparsification algorithm unless NP ⊆ coNP/poly. Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi-arc graphs.
@InProceedings{chen_et_al:LIPIcs.ISAAC.2020.58, author = {Chen, Hubie and Jansen, Bart M. P. and Okrasa, Karolina and Pieterse, Astrid and Rz\k{a}\.{z}ewski, Pawe{\l}}, title = {{Sparsification Lower Bounds for List H-Coloring}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {58:1--58:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.58}, URN = {urn:nbn:de:0030-drops-134027}, doi = {10.4230/LIPIcs.ISAAC.2020.58}, annote = {Keywords: List H-Coloring, Sparsification, Constraint Satisfaction Problem} }
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