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|} (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 ∈ double-struck 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 ⊆ V (G), a list-coloring L : K → 2^{{1, 2, . . ., k∗}} for some k∗ ∈ double-struck N and a parameter k ∈ double-struck 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 2^kn^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).