Random near-regular graphs and the node packing problem

Abstract

Nemhauser and Trotter [12] proposed a certain easily-solved linear program as a relaxation of the node packing problem. They showed that any variables receiving integer values in an optimal solution to this linear program also take on the same values in an optimal solution to the (integer) node packing problem. Let π be the property of graphs defined as follows: a graph G has property π if and only if there is a unique optimal solution to the linear-relaxation problem, and this solution is completely fractional. If a graph G has property π then no information about the node packing problem on G is gained by solving the linear relaxation. We calculate the asymptotic probability that a certain type of ‘sparse’ random graph has property π, as the number of its nodes tends to infinity. Let m be a fixed positive integer, and consider the following random graph on the node set {1,2 …, n}). We join each node, j say, to exactly m other nodes chosen randomly with replacement, by edges oriented away from j; we denote by G n ( m) the undirected graph obtained by deleting all orientations and allowing all parallel edges to coalesce. We show that, as n → ∞, P(G n(m) has property π)→ 0 if m = 1, 1 if m ⩾ 3, and we conjecture that P(G n(2) has property π)→ (1–2e −2) 1 2 .