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Algorithmica

, Volume 81, Issue 3, pp 1267–1287 | Cite as

Parameterized Algorithms for List K-Cycle

  • Fahad PanolanEmail author
  • Saket Saurabh
  • Meirav Zehavi
Article
  • 68 Downloads

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 \(K\subseteq V(G)\) and an injective coloring \(c: K\rightarrow \{1,2,\ldots ,|K|\}\), decide if G has a simple cycle containing each color in \(\{1,2,\ldots ,|K|\}\) exactly once. Another problem widely known since the introduction of color coding is Colorful Cycle. Given a graph G and a coloring \(c: V(G)\rightarrow \{1,2,\ldots ,k\}\) for some \(k\in \mathbb {N}\), it asks if G has a simple cycle of length k containing each color in \(\{1,2,\ldots ,k\}\) exactly once. We study a generalization of these problems: Given a graph G, a set \(K\subseteq V(G)\), a list-coloring \(L: K\rightarrow 2^{\{1,2,\ldots ,k^*\}}\) for some \(k^*\in \mathbb {N}\) and a parameter \(k\in \mathbb {N}\), List\(K\)-Cycle asks if one can assign a color to each vertex in K so that G has 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 \(2^kn^{{{\mathcal {O}}}(1)}\) on an n-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 Björklund, Husfeldt and Taslaman (SODA’12), Björklund, Kaski and Kowalik (STACS’13), and Björklund (FOCS’10).

Keywords

Parameterized complexity K-cycle Colorful path k-path 

Notes

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Authors and Affiliations

  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.The Institute of Mathematical ScienceHBNIChennaiIndia

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