Parameterized Algorithms for List K-Cycle

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 TM V (G). In terms of colored graphs, it can be rephrased as follows: Given a graph G, a set K TM V (G) and an injective coloring c : K æ {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 : V (G) æ {1, 2, . . . , k} for some k œ N, 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 TM V (G), a list-coloring L : K æ 2{1,2,...,k} for some kú œ N and a parameter k œ N, 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 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 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). 1998 ACM Subject Classification G.2.2 Graph Algorithms, I.1.2 Analysis of Algorithms