Exact Exponential Algorithms to Find a Tropical Connected Set of Minimum Size

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 \(\mathcal {O}^*(1.5359^n)\) time algorithm for general graphs and an \(\mathcal {O}^*(1.2721^n)\) time algorithm for trees.

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