Enumerating global roundings of an outerplanar graph

Given a connected weighted graph G = (V, E), we consider a hypergraph H _G = (V, 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 H _G is a binary assignment satisfying that |∑ _v∈Fa(v)-α(v)| < 1 for every F ∈ P _G. Asano et al [1] conjectured that there are at most |V|+1 global roundings for 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.