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11
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 126 (9 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
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 55 (4 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...
A Simple Combinatorial Proof of Duality of Multiroute Flows and Cuts
, 2004
"... A classical ow is a nonnegative linear combination of unit ows along simple paths. A multiroute ow, rst considered by Kishimoto and Takeuchi, generalizes this concept. The basic building blocks are not single paths with unit ows but rather tuples consisting of k edge disjoint paths, each path with ..."
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Cited by 8 (3 self)
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A classical ow is a nonnegative linear combination of unit ows along simple paths. A multiroute ow, rst considered by Kishimoto and Takeuchi, generalizes this concept. The basic building blocks are not single paths with unit ows but rather tuples consisting of k edge disjoint paths, each path with a unit ow. A multiroute ow is a nonnegative linear combination of such tuples.
The Maximum Flow Algorithm Applied to the Placement and Distributed SteadyState Control of FACTS Devices
 Control of FACTS Devices, Proceedings of the 2005 North American Power Symposium
, 2005
"... The bulk power system is one of the largest manmade networks and its size makes control an extremely difficult task. This paper presents a method to control a power network using UPFCs set to levels determined by a maximum flow (maxflow) algorithm. The graphtheorybased maxflow is applied to the p ..."
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Cited by 6 (1 self)
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The bulk power system is one of the largest manmade networks and its size makes control an extremely difficult task. This paper presents a method to control a power network using UPFCs set to levels determined by a maximum flow (maxflow) algorithm. The graphtheorybased maxflow is applied to the power system for UPFC placement and scheduling. A distributed version of maxflow is described to coordinate the actions of the UPFCs distributed in a power network. Two sample power systems were tested using maxflow for UPFC placement and settings. The resulting system characteristics are examined over all singleline contingencies and the appropriateness of the maximum flow algorithm for power flow control is discussed.
Linear Discrepancy of Totally Unimodular Matrices
 Combinatorica
, 2001
"... We show that the linear discrepancy of a totally unimodular mn matrix A is at most lindisc(A) 1 1 n+1 : This bound is sharp. In particular, this result proves Spencer's conjecture lindisc(A) (1 1 n+1 ) herdisc(A) in the special case of totally unimodular matrices. If m 2, we also sho ..."
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Cited by 5 (3 self)
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We show that the linear discrepancy of a totally unimodular mn matrix A is at most lindisc(A) 1 1 n+1 : This bound is sharp. In particular, this result proves Spencer's conjecture lindisc(A) (1 1 n+1 ) herdisc(A) in the special case of totally unimodular matrices. If m 2, we also show lindisc(A) 1 1 m . Finally we give a characterization of those totally unimodular matrices which have linear discrepancy 1 1 n+1 : Besides m 1 matrices containing a single nonzero entry, they are exactly the ones which contain n + 1 rows such that each n thereof are linearly independent. A central proof idea is the use of linear programs. A preliminary version of this result appeared at SODA 2001. This work was partially supported by the graduate school `Eziente Algorithmen und Multiskalenmethoden', Deutsche Forschungsgemeinschaft y A similar result has been independently obtained by T. Bohman and R. Holzman and presented at the Conference on Hypergraphs (Gyula O.H. Katona is 60), Budapest, in June 2001. Mathematics Subject Classication (2000): Primary 11K38, 90C05. Secondary 05C65. Proposed abbreviated title: Linear Discrepancy. 2 1
Optimal MinimumSurface Computations Using Network Flow
"... We give a computationallyefficient solution to a discrete version of the "Plateau problem" on minimal surfaces. Our approach is based on a novel transformation using network flows to find minimumcost slabs, which correspond to minimal "surfaces" of prescribed thickness. An i ..."
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Cited by 2 (0 self)
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We give a computationallyefficient solution to a discrete version of the "Plateau problem" on minimal surfaces. Our approach is based on a novel transformation using network flows to find minimumcost slabs, which correspond to minimal "surfaces" of prescribed thickness. An implementation confirmed that this approach is viable for computing minimal surface solutions for a variety of problem instances.
The Maximum Flow Algorithm Applied to the Placement and Distributed SteadyState
 Control of FACTS Devices, Proceedings of the 2005 North American Power Symposium
, 2005
"... AbstractThe bulk power system is one of the largest manmade networks and its size makes control an extremely difficult task. This paper presents a method to control a power network using UPFCs set to levels determined by a maximum flow (maxflow) algorithm. The graphtheorybased maxflow is applied ..."
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Cited by 1 (1 self)
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AbstractThe bulk power system is one of the largest manmade networks and its size makes control an extremely difficult task. This paper presents a method to control a power network using UPFCs set to levels determined by a maximum flow (maxflow) algorithm. The graphtheorybased maxflow is applied to the power system for UPFC placement and scheduling. A distributed version of maxflow is described to coordinate the actions of the UPFCs distributed in a power network. Two sample power systems were tested using maxflow for UPFC placement and settings. The resulting system characteristics are examined over all singleline contingencies and the appropriateness of the maximum flow algorithm for power flow control is discussed. Index Termsload flow control, operations research, power transmission control, power system control, unified power flow control I.
An Invitation to Sample Paths of Brownian Motion
, 2000
"... 2Preface. These notes record lectures I gave at the Statistics Department, University of California, Berkeley in Spring 1998. I am grateful to the students who attended the course and wrote the rst draft of the notes: Diego Garcia, Yoram Gat, Diogo A. Gomes, Charles Holton, Frederic Latremoliere, We ..."
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Cited by 1 (0 self)
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2Preface. These notes record lectures I gave at the Statistics Department, University of California, Berkeley in Spring 1998. I am grateful to the students who attended the course and wrote the rst draft of the notes: Diego Garcia, Yoram Gat, Diogo A. Gomes, Charles Holton, Frederic Latremoliere, Wei Li, Ben Morris, Jason Schweinsberg, Balint Virag, Ye Xia and Xiaowen Zhou. The draft was edited by Balint Virag and Elchanan Mossel. I thank Pertti Mattila for the invitation to lecture on this material at the joint summer school in
Welfare and Equity Consequences of Transplant Organ Allocation Policies
"... Within the last decade kidney exchange became a mainstream paradigm to increase the number of kidneys transplants. However, compatible pairs do not participate in exchange and full benet from exchange can only be realized if they participate. In this paper, we propose a new incentive scheme that rel ..."
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Within the last decade kidney exchange became a mainstream paradigm to increase the number of kidneys transplants. However, compatible pairs do not participate in exchange and full benet from exchange can only be realized if they participate. In this paper, we propose a new incentive scheme that relies on incentivizing participation of compatible pairs in exchange via an insurance for the patient for a second future renal failure. Welfare and equity analysis of this scheme is conducted and compared with welfare and equity outcomes of live donation and live donor organ exchange. The potential role of such an incentive scheme to strengthen the national kidney exchange system is also presented. M. Utku Unver acknowledges the research support of NSF. We thank the participants at NBER Market Design