Exact Exponential Algorithms to Find a Tropical Connected Set of Minimum Size
Résumé
The input of the Tropical Connected Set problem is a vertex-colored graph G=(V,E) and the task is to find a connected subset $S\subseteq V$ 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 show that Tropical Connected Set on trees has no subexponential-time algorithm unless the Exponential Time Hypothesis fails. This motivates the study of 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.