Enumerating Global Roundings of an Outerplanar Graph

Abstract

Given a connected weighted graph G = (V, E), we consider a hypergraph $\mathcal{H}_G = (V,\mathcal{P}_G)$ corresponding to the set of all shortest paths in G. For a given real assignment a on V satisfying 0 ≤ a(v) ≤ 1, a global rounding α with respect to $\mathcal{H}_G$ is a binary assignment satisfying that | ∑  _v ∈ F a(v) − α(v)| < 1 for every $F \in \mathcal{P}_G$. Asano et al [1] conjectured that there are at most |V|+1 global roundings for $\mathcal{H}_G$. In this paper, we prove that the conjecture holds if G is an outerplanar graph. Moreover, we give a polynomial time algorithm for enumerating all the global roundings of an outerplanar graph.