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### Parameterized Algorithms for List K-Cycle

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### Abstract

The classic K-Cycle problem asks if a graph G, with vertex set V(G), has a simple cycle containing all vertices of a given set K subseteq V(G). In terms of colored graphs, it can be rephrased as follows: Given a graph G, a set K subset of V(G) and an injective coloring c from K to {1,2,...,|K|}, decide if G has a simple cycle containing each color in {1,2,...,|K|} (once). Another problem widely known since the introduction of color coding is {Colorful Cycle}. Given a graph G and a coloring c from V(G) to {1,2,...,k} for some natural number k, it asks if G has a simple cycle of length k containing each color in {1,2,...,k} (once). We study a generalization of these problems: Given a graph G, a set K subset of V(G), a list-coloring L from K to 2^{{1,2,...,k^*}} for some natural number k^* and a parameter k, List K-Cycle asks if one can assign a color to each vertex in K so that G would have a simple cycle (of arbitrary length) containing exactly k vertices from K with distinct colors.

We design a randomized algorithm for List K-Cycle running in time 2^kn^{O(1)} on an -vertex graph, matching the best known running times of algorithms for both K-Cycle and Colorful Cycle. Moreover, unless the Set Cover Conjecture is false, our algorithm is essentially optimal. We also study a variant of List K-Cycle that generalizes the classic Hamiltonicity problem, where one specifies the size of a solution. Our results integrate three related algebraic approaches, introduced by Bjorklund, Husfeldt and Taslaman (SODA'12), Bjorklund, Kaski and Kowalik (STACS'13), and Bjorklund (FOCS'10).

### BibTeX - Entry

```@InProceedings{panolan_et_al:LIPIcs:2016:6857,
author =	{Fahad Panolan and Meirav Zehavi},
title =	{{Parameterized Algorithms for List K-Cycle}},
booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
pages =	{22:1--22:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-027-9},
ISSN =	{1868-8969},
year =	{2016},
volume =	{65},
editor =	{Akash Lal and S. Akshay and Saket Saurabh and Sandeep Sen},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
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