eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-12-10
22:1
22:15
10.4230/LIPIcs.FSTTCS.2016.22
article
Parameterized Algorithms for List K-Cycle
Panolan, Fahad
Zehavi, Meirav
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).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol065-fsttcs2016/LIPIcs.FSTTCS.2016.22/LIPIcs.FSTTCS.2016.22.pdf
Parameterized Complexity
K-Cycle
Colorful Path
k-Path