Average Case Polyhedral Complexity of the Maximum Stable Set Problem

We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via LPs (more precisely, linear extended formulations), in two distinct models. In the uniform model, the constraints of the LP are not allowed to depend on the input graph, which should be encoded solely in the objective function. There we prove a 2Ω(n/ log n) lower bound with probability at least 1− 2−2 for every LP that is exact for a randomly selected set of instances; each graph on at most n vertices being selected independently with probability p > 2−( n/4 2 )+n. In the non-uniform model, the constraints of the LP may depend on the input graph, but we allow weights on the vertices. The input graph is sampled according to the G(n, p) model. There we obtain upper and lower bounds holding with high probability for various ranges of p. The bounds are close as there is only an essentially quadratic gap in the exponent. Finally, we state a conjecture to close the gap.