Exact exponential algorithms to find tropical connected sets of minimum size

Tropical Connected Set is strongly related to the Graph Motif problem which deals with vertex-colored graphs. Graph Motif has various applications in biology and metabolic networks, and has widely been studied in the last twenty years.The input of the Tropical Connected Set problem is a vertex-colored graph (G,c), where G=(V,E) is a graph and c is a vertex coloring assigning to each vertex of G a color. The task is to find a connected subset SV of minimum size such that each color of G appears in S. This problem is known to be NP-complete, even when restricted to trees of height at most three. We study exact exponential algorithms to solve Tropical Connected Set. We present an O(1.5359n) time algorithm for general graphs and an O(1.2721n) time algorithm for trees. We also show that Tropical Connected Set on trees has no sub-exponential algorithm unless the Exponential Time Hypothesis fails.