Deterministic parameterized algorithms for the graph motif problem

We study the classic Graph Motif problem: given a graph G=(V,E) with a set of colors for each node, and a multiset M of colors, we seek a subtree T⊆G, and a coloring of the nodes in T, such that T carries exactly (also with respect to multiplicity) the colors in M. Graph Motif plays a central role in the study of pattern matching problems, primarily motivated from the analysis of complex biological networks. Previous algorithms for Graph Motif and its variants either rely on techniques for developing randomized algorithms that, if derandomized, render them inefficient, or the algebraic narrow sieves technique for which there is no known derandomization. In this paper, we present fast deterministic parameterized algorithms for Graph Motif and its variants. Specifically, we give such an algorithm for the more general Graph Motif with Deletions problem, followed by faster algorithms for Graph Motif and other well-studied special cases. Our algorithms make non-trivial use of representative families, and a novel tool that we call guiding trees, together enabling the efficient construction of the output tree.