Abstract
Given a connected weighted graph G=(V,E), we consider a hypergraph H(G)=(V,F(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∈F(G). Asano et al. [3] conjectured that there are at most |V|+1 global roundings for H(G). In this paper, we present monotone properties on size and affine corank of a set of global roundings under a clique connection operation.
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