Heuristics for Semirandom Graph Problems

摘要

We consider semirandom graph models for finding large independent sets, colorings, and bisections in graphs. These models generate problem instances by blending random and adversarial decisions. To generate semirandom independent set problems, an independent set S of αn vertices is randomly chosen. Each edge connecting S with S is chosen with probability p, and an adversary is then allowed to add new edges arbitrarily, provided that S remains an independent set. The smaller p is, the greater the control the adversary has over the semirandom graph. We give a heuristic that with high probability recovers an independent set of size αn whenever p> (1+ε) ln n/αn, for any constant ε>0. We show that when p<(1−ε) ln n /αn, an independent set of size |S| cannot be recovered, unless NP⊆BPP. We use our result for maximum independent sets to obtain greatly improved heuristics for the model of k-colorable semirandom graphs introduced by Blum and Spencer. For constant k, our results are optimal up to constant factors in the edge probabilities. In the semirandom model for graph bisection, a random bisection (S, S) of the vertices is chosen. Each edge (u, v)∈S×S is independently chosen with probability q and each edge (u, v)∉S×S is independently chosen with probability p>q. The adversary may then arbitrarily remove edges in S×S and add edges not in S×S. Extending the work of Boppana, we give a heuristic that recovers this bisection with high probability when p−q⩾c , for c a sufficiently large constant.