Exact and parameterized algorithms for choosability

In the C hoosability problem (or list chromatic number problem), for a given graph G , we need to find the smallest k such that G admits a list coloring for any list assignment where all lists contain at least k colors. The problem is tightly connected with the well-studied C oloring and L ist C olo...

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Vydáno v:Acta informatica Ročník 62; číslo 2; s. 24
Hlavní autoři: Bliznets, Ivan, Nederlof, Jesper
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2025
Springer Nature B.V
Témata:
Algorithms
Apexes
Coloring
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Information Systems and Communication Service
Logics and Meanings of Programs
Lower bounds
Parameterization
Polynomials
Software Engineering/Programming and Operating Systems
Theory of Computation
Upper bounds
ISSN:0001-5903, 1432-0525
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Shrnutí:In the C hoosability problem (or list chromatic number problem), for a given graph G , we need to find the smallest k such that G admits a list coloring for any list assignment where all lists contain at least k colors. The problem is tightly connected with the well-studied C oloring and L ist C oloring problems. However, the knowledge of the complexity landscape for the C hoosability problem is pretty scarce. Moreover, most of the known results only provide lower bounds for its computational complexity and do not provide ways to cope with the intractability. The main objective of our paper is to construct the first non-trivial exact exponential algorithms for the C hoosability problem, and complete the picture with parameterized results. Specifically, we present the first single-exponential algorithm for the decision version of the problem with fixed k . This result answers an implicit question from Eppstein on a stackexchange thread discussing upper bounds on the union of lists assigned to vertices. We also present a time algorithm for the general C hoosability problem. In the parameterized setting, we give a polynomial kernel for the problem parameterized by vertex cover, and algorithms that run in FPT time when parameterized by a size of a clique-modulator and by the dual parameterization . Additionally, we show that C hoosability admits a significant running time improvement if it is parameterized by cutwidth in comparison with the parameterization by treewidth studied by Marx and Mitsou [ICALP’16]. On the negative side, we provide a lower bound parameterized by a size of a modulator to split graphs under assumption of the Exponential Time Hypothesis.
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ISSN:0001-5903
1432-0525
DOI:10.1007/s00236-025-00492-0