Large Hinge Width on Sparse Random Hypergraphs
Tian Liu, Xiaxiang Lin, Chaoyi Wang, Kaile Su, Ke Xu
Consider random hypergraphs on n vertices, where each k-element subset of vertices is selected with probability p independently and randomly as a hyperedge. By sparse we mean that the total number of hyperedges is O(n) or O(n ln n). When k = 2, these are exactly the classical Erdös-Rényi random graphs G(n,p). We prove that with high probability, hinge width on these sparse random hypergraphs can grow linearly with the expected number of hyperedges. Some random constraint satisfaction problems such as Model RB and Model RD have satisfiability thresholds on these sparse constraint hypergraphs, thus the large hinge width results provide some theoretical evidence for random instances around satisfiability thresholds to be hard for a standard hinge-decomposition based algorithm. We also conduct experiments on these and other kinds of random graphs with several hundreds vertices, including regular random graphs and power law random graphs. The experimental results also show that hinge width can grow linearly with the number of edges on these different random graphs. These results may be of further interests.