Experimental Evaluation of the Greedy and Random Algorithms for Finding Independent Sets in Random Graphs

Abstract

This work is motivated by the long-standing open problem of designing a polynomial-time algorithm that with high probability constructs an asymptotically maximum independent set in a random graph. We present the results of an experimental investigation of the comparative performance of several efficient heuristics for constructing maximal independent sets. Among the algorithms that we evaluate are the well known randomized heuristic, the greedy heuristic, and a modification of the latter which breaks ties in a novel way. All algorithms deliver on-line upper bounds on the size of the maximum independent set for the specific input-graph. In our experiments, we consider random graphs parameterized by the number of vertices n and the average vertex degree d. Our results provide strong experimental evidence in support of the following conjectures: for d = c · n (c is a constant), the greedy and random algorithms are asymptotically equivalent; for fixed d, the greedy algorithms are asymptotically superior to the random algorithm; for graphs with d ≤ 3, the approximation ratio of the modified greedy algorithm is asymptotically < 1.005. We also consider random 3-regular graphs, for which non-trivial lower and upper bounds on the size of a maximum independent set are known. Our experiments suggest that the lower bound is asymptotically tight.