Exact and parameterized algorithms for choosability.

In the Choosability 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 Coloring and List Coloring problems. However, the knowledge of the complexity landscape for the Choosability 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 Choosability 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 [Formula: see text] time algorithm for the general Choosability 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 [Formula: see text]. Additionally, we show that Choosability 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.