Problems such as bisection, graph coloring, and clique are generally believed hard in the worst case. However, they can be solved if the input data is drawn randomly from a distribution over graphs containing acceptable solutions. In this paper we show that a simple spectral algorithm can solve all three problems above in the average case, as well as a more general problem of partitioning graphs based on edge density. In nearly all cases our approach meets or exceeds previous parameters, while introducing substantial generality. We apply spectral techniques, using foremost the observation that in all of these problems, the expected adjacency matrix is a low rank matrix wherein the structure of the solution is evident.

We resolve in the affirmative a question of Boppana and Bui: whether simulated annealing can, with high probability and in polynomial time, find the optimal bisection of a random graph in Gnpr when p- r = O(n*-’) for A 5 2. (The random graph model Gnpr specifies a “planted ” bisection of density r, separating two n/2-vertex subsets of slightly higher density p.) We show that simulated “annealing ” at an appropriate fixed temperature (i.e., the Metropolis algorithm) finds the unique smallest bisection in O(n2+‘) steps with very high probability, provided A> 1116. (By using a slightly modified neighborhood structure, the number of steps can be reduced to O(n’+‘).) We leave open the question of whether annealing is effective for A in the range 312 < A 5 1116, whose lower limit represents the threshold at which the planted bisection becomes lost amongst other random small bisections. It also remains open whether hillclimbing (i.e., annealing at temperature 0) solves the same problem. 1

We analyze the behavior of hill-climbing algorithms for the minimum bisection problem on instances drawn from the "planted bisection" random graph model, Gn;p;q , previously studied in [3, 4, 10, 12, 15, 9, 7]. This is one of the few problem distributions for which various popular heuristic methods, such as simulated annealing, have been proven to succeed. However, it has been open whether these sophisticated methods were necessary, or whether simpler heuristics would also work. Juels [15] made the first progress towards an answer by showing that simple hill-climbing does suffice for very wide separations between p and q.

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Abstract. Random 3-colorable graphs that are generated according to a G(n, p)-like model can be colored optimally, if p ≥ c/n for some large constant c. However, these methods fail in a model where the edgeprobabilities are non-uniform and not bounded away from zero. We present a spectral algorithm that succeeds in such situations. 1

Random 3-colorable graphs that are generated according to a G(n, p)-like model can be colored optimally, if p ≥ c/n for some large constant c. However, these methods fail in a model where the edgeprobabilities are non-uniform and not bounded away from zero. We present a spectral algorithm that succeeds in such situations.