The structure and number of global roundings of a graph

@article{Asano2004TheSA,
  title={The structure and number of global roundings of a graph},
  author={Tetsuo Asano and Naoki Katoh and Hisao Tamaki and Takeshi Tokuyama},
  journal={Theor. Comput. Sci.},
  year={2004},
  volume={325},
  pages={425-437}
}
Given a connected weighted graph G = (V,E), we consider a hypergraph HG = (V, PG) 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 HG is a binary assignment satisfying that |Σv∈Fa(v)-α(v)| <1 for every F ∈ PG. We conjecture that there are at most |V| + 1 global roundings for HG, and also the set of global roundings is an affine independent set. We give several positive evidences for the conjecture. 
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