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Spectral Partitioning of Random Graphs
, 2001
"... 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 ..."
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Cited by 86 (3 self)
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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.
Simulated annealing for graph bisection
 in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science
, 1993
"... 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, ..."
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Cited by 34 (1 self)
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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/2vertex 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
HillClimbing Finds Random Planted Bisections
 Proc. 12th Symposium on Discrete Algorithms (SODA 01), ACM press, 2001
, 2001
"... We analyze the behavior of hillclimbing 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, ..."
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Cited by 11 (1 self)
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We analyze the behavior of hillclimbing 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 hillclimbing does suffice for very wide separations between p and q.
Coloring Random 3Colorable Graphs with Nonuniform Edge Probabilities ⋆
"... Abstract. Random 3colorable 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 nonuniform and not bounded away from zero. We present a spectral algorithm ..."
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Cited by 1 (1 self)
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Abstract. Random 3colorable 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 nonuniform and not bounded away from zero. We present a spectral algorithm that succeeds in such situations. 1
Coloring Random 3Colorable Graphs with Nonuniform Edge Probabilities
, 2006
"... Random 3colorable 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 nonuniform and not bounded away from zero. We present a spectral algorithm that succ ..."
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Cited by 1 (1 self)
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Random 3colorable 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 nonuniform and not bounded away from zero. We present a spectral algorithm that succeeds in such situations.
Are Stable Instances Easy? ∗
"... Abstract: We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP–hard problems are easier to solve. In particular, whethe ..."
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Abstract: We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP–hard problems are easier to solve. In particular, whether there exist algorithms that solve correctly and in polynomial time all sufficiently stable instances of some NP–hard problem. The paper focuses on the Max–Cut problem, for which we show that this is indeed the case.
doi:10.1017/S0963548312000193 Are Stable Instances Easy?
, 2009
"... We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NPhard problems are easier to solve, and in particular, whether ther ..."
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We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NPhard problems are easier to solve, and in particular, whether there exist algorithms that solve in polynomial time all sufficiently stable instances of some NPhard problem. The paper focuses on the MaxCut problem, for which we show that this is indeed the case.