Results 1  10
of
14
A new approach to the minimum cut problem
 Journal of the ACM
, 1996
"... Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds th ..."
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Cited by 99 (8 self)
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Abstract. This paper presents a new approach to finding minimum cuts in undirected graphs. The fundamental principle is simple: the edges in a graph’s minimum cut form an extremely small fraction of the graph’s edges. Using this idea, we give a randomized, strongly polynomial algorithm that finds the minimum cut in an arbitrarily weighted undirected graph with high probability. The algorithm runs in O(n 2 log 3 n) time, a significant improvement over the previous Õ(mn) time bounds based on maximum flows. It is simple and intuitive and uses no complex data structures. Our algorithm can be parallelized to run in �� � with n 2 processors; this gives the first proof that the minimum cut problem can be solved in ���. The algorithm does more than find a single minimum cut; it finds all of them. With minor modifications, our algorithm solves two other problems of interest. Our algorithm finds all cuts with value within a multiplicative factor of � of the minimum cut’s in expected Õ(n 2 � ) time, or in �� � with n 2 � processors. The problem of finding a minimum multiway cut of a graph into r pieces is solved in expected Õ(n 2(r�1) ) time, or in �� � with n 2(r�1) processors. The “trace ” of the algorithm’s execution on these two problems forms a new compact data structure for representing all small cuts and all multiway cuts in a graph. This data structure can be efficiently transformed into the
Minimum Cuts in NearLinear Time
, 1999
"... We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorit ..."
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Cited by 73 (11 self)
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We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorithm that finds a minimum cut in an medge, nvertex graph with high probability in O(m log³ n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n² log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n² log³ n). Other applications of the treepacking approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a spaceefficient manner.
An NC Algorithm for Minimum Cuts
 IN PROCEEDINGS OF THE 25TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
"... We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from ..."
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Cited by 48 (3 self)
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We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from the minimum cut problem to the problem of obtaining a (2 + ffl)approximation to the minimum cut. This reduction involves a natural combinatorial SetIsolation Problem that can be solved easily in RNC. The third result is a derandomization of this RNC solution that requires a combination of two widely used tools: pairwise independence and random walks on expanders. We believe that the setisolation approach will prove useful in other derandomization problems. The techniques extend to two related problems: we describe NC algorithms finding minimum kway cuts for any constant k and finding all cuts of value within any constant factor of the minimum. Another application of these techni...
Experimental Study of Minimum Cut Algorithms
 PROCEEDINGS OF THE EIGHTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA)
, 1997
"... Recently, several new algorithms have been developed for the minimum cut problem. These algorithms are very different from the earlier ones and from each other and substantially improve worstcase time bounds for the problem. We conduct experimental evaluation the relative performance of these algor ..."
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Cited by 40 (2 self)
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Recently, several new algorithms have been developed for the minimum cut problem. These algorithms are very different from the earlier ones and from each other and substantially improve worstcase time bounds for the problem. We conduct experimental evaluation the relative performance of these algorithms. In the process, we develop heuristics and data structures that substantially improve practical performance of the algorithms. We also develop problem families for testing minimum cut algorithms. Our work leads to a better understanding of practical performance of the minimum cut algorithms and produces very efficient codes for the problem.
Compiling a Massive, Multilingual Dictionary via Probabilistic Inference
, 2009
"... Can we automatically compose a large set of Wiktionaries and translation dictionaries to yield a massive, multilingual dictionary whose coverage is substantially greater than that of any of its constituent dictionaries? The composition of multiple translation dictionaries leads to a transitive infer ..."
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Cited by 13 (1 self)
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Can we automatically compose a large set of Wiktionaries and translation dictionaries to yield a massive, multilingual dictionary whose coverage is substantially greater than that of any of its constituent dictionaries? The composition of multiple translation dictionaries leads to a transitive inference problem: if word A translates to word B which in turn translates to word C, what is the probability that C is a translation of A? The paper introduces a novel algorithm that solves this problem for 10,000,000 words in more than 1,000 languages. The algorithm yields PANDICTIONARY, a novel multilingual dictionary. PANDICTIONARY contains more than four times as many translations than in the largest Wiktionary at precision 0.90 and over 200,000,000 pairwise translations in over 200,000 language pairs at precision 0.8.
Using random sampling to find maximum flows in uncapacitated undirected graphs
 In Proceedings of the 29th ACM Symposium on Theory of Computing
, 1997
"... vertex capacitated graphs. However, these schemes could not find flows or exactstmincuts, even in uncapacitated graphs. We present new algorithms, based on random sampling, that In this work, we show that sampling can speed up the find maximum flows in undirected uncapacitated graphs. Our fastest ..."
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Cited by 10 (4 self)
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vertex capacitated graphs. However, these schemes could not find flows or exactstmincuts, even in uncapacitated graphs. We present new algorithms, based on random sampling, that In this work, we show that sampling can speed up the find maximum flows in undirected uncapacitated graphs. Our fastest known algorithms for findingstmaxflows (and thus algorithms dominate augmenting paths over all parameter val exactstmincuts) in uncapacitated undirected graphs. Until ues (number of vertices and edges and flow value). They also now, the best algorithms have used augmenting paths or blockdominate blocking flows over a large range of parameter valing flows [Tar83, AMO93]. Augmenting paths can be used to ues. Furthermore, they achieve time bounds on graphs with find a flow of valuevinO(mv)=O(n3)time on anmparallel (equivalently, capacitated) edges that previously could edge,nvertex graph. Blocking flows can be used to find a only be achieved on graphs without them. maxflow inO(m3=2)time [Eve79]. Thus augmenting paths The key contribution of this paper is to demonstrate that dominate for small values ofv, while blocking flows dominate such an improvement is possible. This shows that augment whenvis large (greater thanpm). If we restrict the graph to ing paths and blocking flows are nonoptimal, and reopens the have no parallel edges, blocking flows achieve a time bound of
Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs
, 2011
"... We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a nearlineartime randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)edge graph on the same vertices whose cuts have approximately t ..."
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Cited by 9 (0 self)
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We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a nearlineartime randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)edge graph on the same vertices whose cuts have approximately the same value as the original graph’s. In this new graph, for example, we can run the Õ(m3/2)time maximum flow algorithm of Goldberg and Rao to find an s– t minimum cut in Õ(n3/2) time. This corresponds to a (1 + ɛ)times minimum s–t cut in the original graph. A related approach leads to a randomized divide and conquer algorithm producing an approximately maximum flow in Õ(m √ n) time. Our algorithm is also used to improve the running time of sparsest cut algorithms from Õ(mn) to Õ(n²). Our approach also accelerates several other recent cut and flow algorithms. Our algorithms are based on a general theorem analyzing the concentration of cut values near their expectation in random graphs.
Graph Layout Problems Parameterized by Vertex Cover
"... In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are ..."
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Cited by 8 (3 self)
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In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are a notable exception. In particular, no fixed parameter tractable algorithms are known for the Cutwidth, Bandwidth, Imbalance and Distortion problems parameterized by treewidth. In fact, Bandwidth remains NPcomplete even restricted to trees. A possible way to attack graph layout problems is to consider structural parameterizations that are stronger than treewidth. In this paper we study graph layout problems parameterized by the size of the minimum vertex cover of the input graph. We show that all the mentioned problems are fixed parameter tractable. Our basic ingredient is a classical algorithm for Integer Linear Programming when parameterized by dimension, designed by Lenstra and later improved by Kannan. We hope that our results will serve to reemphasize the importance and utility of this algorithm.
Randomization in Graph Optimization Problems: A Survey
 OPTIMA
, 1998
"... Randomization has become a pervasive technique in combinatorial optimization. We survey our thesis and subsequent work, which uses four common randomization techniques to attack numerous optimization problems on undirected graphs. ..."
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Cited by 2 (0 self)
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Randomization has become a pervasive technique in combinatorial optimization. We survey our thesis and subsequent work, which uses four common randomization techniques to attack numerous optimization problems on undirected graphs.