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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 90 (2 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 35 (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 heuristi ..."
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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
, 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.
Spectral Methods for Bisecting Graphs
, 1993
"... this paper have found widespread use in this application [24, 30]. Given the utility of graph bisection, it is disappointing (though not surprising) that this problem is difficult from a complexity pointofview. For general graphs, Garey, Johnson, and Stockmeyer [21] (see also [10]) have shown that ..."
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this paper have found widespread use in this application [24, 30]. Given the utility of graph bisection, it is disappointing (though not surprising) that this problem is difficult from a complexity pointofview. For general graphs, Garey, Johnson, and Stockmeyer [21] (see also [10]) have shown that the decision problem, "Does graph G have a bisection with fewer than k edges?" is NPcomplete. For planar graphs the complexity is unknown, but Papadimitriou and Sideri [28] conjecture that the problem is still NPcomplete. Polynomial time algorithms for finding optimal bisections are known only for trees, boundedwidth planar graphs, and solid grid graphs [28]. Furthermore, no approximation algorithms are known even for planar graphs. The current stateoftheart consists of heuristics that often perform well in practice [2, 4, 11, 25] and algorithms that have been shown to have good average case behavior over certain probability distributions on graphs [7, 9, 11, 14]. The next section briefly surveys some of these algorithms. The rest of this paper focuses on the spectral methods and is organized as follows. Section 3 covers some preliminary facts concerning the eigenvalues and eigenvectors of real, symmetric matrices. This section concludes with two theorems that play a central role in the two spectral algorithms that we will study. One of these two theorems is a slight generalization of a theorem due to Fan [15] that forms the basis for Section 4's simplified exposition of an algorithm due to Barnes [2]. The other theorem fills in a major hole in Boppana's extended abstract [7]. Boppana's algorithm is the topic of Section 5. We point out a problem with the original algorithm, discuss a remedy proposed by Boppana [8], and then demonstrate the problem with the remedy. Fo...
Subsampled Power Iteration: a New Algorithm for Block Models and Planted CSP’s
"... We present a new algorithm for recovering planted solutions in two wellknown models, the stochastic block model and planted constraint satisfaction problems, via a common generalization in terms of random bipartite graphs. Our algorithm achieves the bestknown bounds for the number of edges needed ..."
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We present a new algorithm for recovering planted solutions in two wellknown models, the stochastic block model and planted constraint satisfaction problems, via a common generalization in terms of random bipartite graphs. Our algorithm achieves the bestknown bounds for the number of edges needed for perfect recovery and its running time is linear in the number of edges used. The time complexity is significantly better than both spectral and SDPbased approaches. The main new features of the algorithm are twofold: (i) the critical use of power iteration with subsampling, which might be of independent interest; its analysis requires keeping track of multiple norms of an evolving solution (ii) it can be implemented statistically, i.e., with very limited access to the input distribution.
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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 ..."
Abstract
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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. 1
Sorting Noisy Data with Partial Information
"... In this paper, we propose two semirandom models for the Minimum Feedback Arc Set Problem and present approximation algorithms for them. In the first model, which we call the Random Edge Flipping model, an instance is generated as follows. We start with an arbitrary acyclic directed graph and then ..."
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In this paper, we propose two semirandom models for the Minimum Feedback Arc Set Problem and present approximation algorithms for them. In the first model, which we call the Random Edge Flipping model, an instance is generated as follows. We start with an arbitrary acyclic directed graph and then randomly flip its edges (the adversary may later unflip some of them). In the second model, which we call the Random Backward Edge model, again we start with an arbitrary acyclic graph but now add new random backward edges (the adversary may delete some of them). For the first model, we give an approximation algorithm that finds a solution of cost (1 + δ) optcost+npolylog n, where optcost is the cost of the optimal solution. For the second model, we give an approximation algorithm that finds a solution of cost O(plantedcost)+npolylog n, where plantedcost is the cost of the planted solution. Additionally, we present an approximation algorithm for semirandom instances of Minimum Directed Balanced Cut.