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# Double-Exponential and Triple-Exponential Bounds for Choosability Problems Parameterized by Treewidth

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Dániel Marx and Valia Mitsou. Double-Exponential and Triple-Exponential Bounds for Choosability Problems Parameterized by Treewidth. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ICALP.2016.28

## Abstract

Choosability, introduced by Erdös, Rubin, and Taylor [Congr. Number. 1979], is a well-studied concept in graph theory: we say that a graph is c-choosable if for any assignment of a list of c colors to each vertex, there is a proper coloring where each vertex uses a color from its list. We study the complexity of deciding choosability on graphs of bounded treewidth. It follows from earlier work that 3-choosability can be decided in time 2^(2^(O(w)))*n^(O(1)) on graphs of treewidth w. We complement this result by a matching lower bound giving evidence that double-exponential dependence on treewidth may be necessary for the problem: we show that an algorithm with running time 2^(2^(o(w)))*n^(O(1)) would violate the Exponential-Time Hypothesis (ETH). We consider also the optimization problem where the task is to delete the minimum number of vertices to make the graph 4-choosable, and demonstrate that dependence on treewidth becomes tripleexponential for this problem: it can be solved in time 2^(2^(2^(O(w))))*n^(O(1)) on graphs of treewidth w, but an algorithm with running time 2^(2^(2^(o(w))))*n^(O(1)) would violate ETH.
##### Keywords
• Parameterized Complexity
• List coloring
• Treewidth
• Lower bounds under ETH

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