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12
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 100 (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
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 41 (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.
Experimental Study of Minimum Cut Algorithms
 M.S. DISSERTATION, MIT
, 1997
"... Recently, several new algorithms have been developed for the minimum cut problem that substantially improve worstcase time bounds for the problem. These algorithms are very different from the earlier ones and from each other. We conduct an experimental evaluation of the relative performance of thes ..."
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Cited by 6 (0 self)
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Recently, several new algorithms have been developed for the minimum cut problem that substantially improve worstcase time bounds for the problem. These algorithms are very different from the earlier ones and from each other. We conduct an experimental evaluation of 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.
Reliability in Layered Networks with Random Link Failures
"... Abstract—We consider network reliability in layered networks where the lower layer experiences random link failures. In layered networks, each failure at the lower layer may lead to multiple failures at the upper layer. We generalize the classical polynomial expression for network reliability to the ..."
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Cited by 5 (0 self)
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Abstract—We consider network reliability in layered networks where the lower layer experiences random link failures. In layered networks, each failure at the lower layer may lead to multiple failures at the upper layer. We generalize the classical polynomial expression for network reliability to the multilayer setting. Using random sampling techniques, we develop polynomial time approximation algorithms for the failure polynomial. Our approach gives an approximate expression for reliability as a function of the link failure probability, eliminating the need to resample for different values of the failure probability. Furthermore, it gives insight on how the routings of the logical topology on the physical topology impact network reliability. We show that maximizing the min cut of the (layered) network maximizes reliability in the low failure probability regime. Based on this observation, we develop algorithms for routing the logical topology to maximize reliability. I.
Counting and sampling minimum (s, t)cuts in weighted planar graphs in polynomial time
"... We give an O(nd + n log n) algorithm computing the number of minimum (s, t)cuts in weighted planar graphs, where n is the number of vertices and d is the length of the shortest st path in the corresponding unweighted graph. Previously, Ball and Provan gave a polynomialtime algorithm for unweighte ..."
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Cited by 3 (1 self)
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We give an O(nd + n log n) algorithm computing the number of minimum (s, t)cuts in weighted planar graphs, where n is the number of vertices and d is the length of the shortest st path in the corresponding unweighted graph. Previously, Ball and Provan gave a polynomialtime algorithm for unweighted planar graphs with both s and t lying on the outer face. Our results hold for all locations of s and t and weighted graphs, and have direct applications in computer vision.
Reliable Assignments of Processors to Tasks and Factoring on Matroids
, 1993
"... In the simple assignment problem, there are n processors, m tasks, and a relation between the processors and tasks; this relation indicates the ability of the processor to perform the task. When the processors fail independently with known probabilities, two performance issues arise. First, with wha ..."
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In the simple assignment problem, there are n processors, m tasks, and a relation between the processors and tasks; this relation indicates the ability of the processor to perform the task. When the processors fail independently with known probabilities, two performance issues arise. First, with what probability can the operating processors all be kept busy? Second, with what probability can the operating processors perform the same number of tasks that all processors could? We formulate these questions on the underlying transversal matroid. We first prove that counting minimum cardinality circuits in this matroid is #Pcomplete, and hence that both questions are also #Pcomplete. Secondly, we devise a factoring algorithm with series and parallel reductions to compute exact solutions of the above problems. We then outline some efficient strategies for bounding the probabilities. 1 Introduction Consider a multiprocessor system having a set C= c 1 ; c 2 ; : : : ; c n of processors availa...
I R I S a
"... : An efficient family of methods to evaluate network reliability is the class of factoring algorithms. Their efficiency is due to the use of reliabilitypreserving reductions (for instance, seriesparallel ones). In this work, we follow a similar approach for the problem of counting the (s; t)mincu ..."
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: An efficient family of methods to evaluate network reliability is the class of factoring algorithms. Their efficiency is due to the use of reliabilitypreserving reductions (for instance, seriesparallel ones). In this work, we follow a similar approach for the problem of counting the (s; t)mincuts of a graph with size i, for all i. In order to take into account these possible reductions, the paper presents a technique based on a factoring formula, similar to the relation of Moskowitz, to count the number of (s; t)mincuts of each size. This formula is applied recursively and various reductions, with computational cost polynomial in the size of the network, are used in order to reduce the size of the studied graph. Keywords: (s; t)minimal cutset, factoring algorithm, reliabilitypreserving reductions (R'esum'e : tsvp) * Email: sbulteau@irisa.fr CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE Centre National de la Recherche Scientifique Institut National de Recherche en Informati...
Chapter 2 ENUMERATING NEARMIN ST CUTS
"... Abstract We develop a factoring (partitioning) algorithm for enumerating nearminimumweight st cuts in directed and undirected graphs, with application to network interdiction. “Nearminimum ” means within a factor of 1+ � of the minimum for some � ≥ 0. The algorithm requires only polynomial work ..."
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Abstract We develop a factoring (partitioning) algorithm for enumerating nearminimumweight st cuts in directed and undirected graphs, with application to network interdiction. “Nearminimum ” means within a factor of 1+ � of the minimum for some � ≥ 0. The algorithm requires only polynomial work per cut enumerated provided that � is sufficiently (not trivially) small, or G has special structure, e.g., G is a complete graph. Computational results demonstrate good empirical efficiency even for large values of � and for general graph topologies.
Experimental Study of Minimum Cut Algorithms
 M.S. dissertation, MIT
, 1995
"... Recently, several new algorithms have been developed for the minimum cut problem that substantially improve worstcase time bounds for the problem. These algorithms are very different from the earlier ones and from each other. We conduct an experimental evaluation of the relative performance of thes ..."
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Recently, several new algorithms have been developed for the minimum cut problem that substantially improve worstcase time bounds for the problem. These algorithms are very different from the earlier ones and from each other. We conduct an experimental evaluation of 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. Keywords: minimum cut, graph algorithms, experimental evaluation, network optimization Thesis Supervisor: David Karger Title: Assistant Professor of Electrical Engineering and Computer Science This research was partly supported by DARPA contracts N0000149511246 and DABT6395C0009, Army Contract DAAH049510607 and NSF Award CCR9624239. Some of this work was done at NEC Research Institute. Contents 1
Counting minimum (s, t)cuts in weighted planar graphs in polynomial time
"... Abstract. We give an O(nd+n log n) algorithm computing the number of minimum (s, t)cuts in weighted planar graphs, where n is the number of vertices and d is the length of the shortest st path in the corresponding unweighted graph. Previously, Ball and Provan gave a polynomialtime algorithm for u ..."
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Abstract. We give an O(nd+n log n) algorithm computing the number of minimum (s, t)cuts in weighted planar graphs, where n is the number of vertices and d is the length of the shortest st path in the corresponding unweighted graph. Previously, Ball and Provan gave a polynomialtime algorithm for unweighted graphs with both s and t lying on the outer face. Our results hold for all locations of s and t and weighted graphs, and have direct applications in image segmentation and other computer vision problems. 1