Parameterized Algorithms for List K-Cycle

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).