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⊆V(G). In terms of colored graphs, it can be rephrased as follows: Given a graph G, a set K⊆V(G) and an injective coloring c:K→{1,2,…,|K|}, decide if G has a simple cycle containing each color in {1,2,…,|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)→{1,2,…,k} for some k∈N, it asks if G has a simple cycle of length k containing each color in {1,2,…,k} exactly once. We study a generalization of these problems: Given a graph G, a set K⊆V(G), a list-coloring L:K→2{1,2,…,k∗} for some k∗∈N and a parameter k∈N, LISTK-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 LISTK-CYCLE running in time 2knO(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 LISTK-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).