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⊆ 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 2 ^k n ^{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 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).