Large Hinge Width on Sparse Random Hypergraphs
Xiaxiang Lin, Tian Liu, Chaoyi Wang, Ke Xu and Yang Yuan
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 Erdos-Renyi 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 have satisfiability thresholds on these sparse constraint hypergraphs, thus the large hinge width results imply that no efficient solver based on hinge decomposition can exist for random instances around satisfiability thresholds. 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.