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16
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
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.
QOS Routing Via Multiple Paths Using Bandwidth Reservation
 In IEEE INFOCOM98: The Conference on Computer Communications
, 1998
"... vii 1 Introduction 1 1.1 Relation to Prior Work : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Contribution and Organization of the Paper : : : : : : : : : : : : : : 3 2 Problem Formulation 4 3 Message Transmission Problem 5 3.1 ShortestWidest Paths : : : : : : : : : : : : : : : : : : : ..."
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Cited by 39 (4 self)
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vii 1 Introduction 1 1.1 Relation to Prior Work : : : : : : : : : : : : : : : : : : : : : : : : : : 2 1.2 Contribution and Organization of the Paper : : : : : : : : : : : : : : 3 2 Problem Formulation 4 3 Message Transmission Problem 5 3.1 ShortestWidest Paths : : : : : : : : : : : : : : : : : : : : : : : : : : 5 3.2 Properties of Multipaths : : : : : : : : : : : : : : : : : : : : : : : : : 6 3.3 NPCompleteness of MTP : : : : : : : : : : : : : : : : : : : : : : : : 10 3.4 Approximate Routing Algorithm : : : : : : : : : : : : : : : : : : : : 14 3.5 Relation to Maximum Flow Algorithm : : : : : : : : : : : : : : : : : 16 3.6 Simulation Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18 3.7 DelayBandwidth Product : : : : : : : : : : : : : : : : : : : : : : : : 24 4 Sequence Transmission Problem 25 4.1 Intractability Results : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 4.2 Approximation Algorithm : : : : : : : : : : : : : : : : : : : : : : : : 28 5 Concl...
Online Load Balancing and Network Flow
 In Proc. 25th Annual ACM Symposium on Theory of Computing
, 1993
"... In this paper we study two problems that can be viewed as online games on a dynamic bipartite graph. The first problem is online load balancing with preemption. A centralized scheduler must assign tasks to servers, processing online a sequence of task arrivals and departures. Each task is restrict ..."
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Cited by 34 (4 self)
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In this paper we study two problems that can be viewed as online games on a dynamic bipartite graph. The first problem is online load balancing with preemption. A centralized scheduler must assign tasks to servers, processing online a sequence of task arrivals and departures. Each task is restricted to run on some subset of the servers. The scheduler attempts to keep the load wellbalanced. If preemptive reassignments are dissallowed, Azar, Broder and Karlin [3] proved a lower bound of \Omega\Gamma p n) on the ratio between the maximum load achieved by an online algorithm and the optimum offline maximum load. We show that this ratio can be greatly reduced by an efficient scheduler using only a small amount of rescheduling. We then apply these ideas to network flow. Cheriyan and Hagerup [6] introduced an online game on a bipartite graph as a fundamental step in improving algorithms for computing the maximum flow in networks. They described a randomized strategy to play the game. ...
Computational aspects of clearing continuous call double auctions with assignment constraints and indivisible demand
 Electronic Commerce Research
, 2000
"... We investigate the computational complexity of clearing markets in a continuous call double auction. In the simplest case, when any part of any bid can be matched with any part of any ask, the market can be cleared optimally in loglinear time. We present two generalizations, motivated by electronic ..."
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Cited by 31 (0 self)
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We investigate the computational complexity of clearing markets in a continuous call double auction. In the simplest case, when any part of any bid can be matched with any part of any ask, the market can be cleared optimally in loglinear time. We present two generalizations, motivated by electronic marketplaces for the process industry, where: (a) there exist assignment constraints on which bids can be matched with which asks; and (b) where demand is indivisible. We show that clearing markets with assignment constraints can be solved efficiently using network flow algorithms. However clearing markets with indivisible demand, with or without assignment constraints, requires solving NPhard optimization problems such as the generalized assignment problem, the multiple knapsack problem and the bin packing problem.
Can a Maximum Flow be Computed in o(nm) Time?
 IN PROC. ICALP
, 1990
"... We show that a maximum flow in a network with n vertices can be computed deterministically in O(n³/log n) time on a uniformcost RAM. For dense graphs, this improves the previous best bound of O(n³). The bottleneck in our algorithm is a combinatorial problem on (unweighted) graphs. The number of op ..."
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Cited by 15 (0 self)
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We show that a maximum flow in a network with n vertices can be computed deterministically in O(n³/log n) time on a uniformcost RAM. For dense graphs, this improves the previous best bound of O(n³). The bottleneck in our algorithm is a combinatorial problem on (unweighted) graphs. The number of operations executed on flow variables is O(nS/3(log n)4/3), in contrast with f~(nm) flow operations for all previous algorithms, where m denotes the number of edges in the network. A randomized version of our algorithm executes O(nal2rn 1/2 (log n) 312 + n 2 (log n) 2) flow operations with high probability. Specializing to the case in which all capacities are integers bounded by U, we show that a maximum flow can be computed using O(n3/2ml/2 + n2{log U) ~/2) flow operations. Finally, we argue that several of our results yield optimal parallel algorithms.
Length Functions for Flow Computations
 NEC Research Institute
, 1997
"... We introduce a new approach to the maximum flow problem. This approach is based on assigning arc lengths based on the residual flow value and the residual arc capacities. Our approach leads to an O(min(n 2=3 ; m 1=2 )m log( n 2 m ) log U) time bound for a network with n vertices, m arcs, and ..."
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Cited by 11 (0 self)
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We introduce a new approach to the maximum flow problem. This approach is based on assigning arc lengths based on the residual flow value and the residual arc capacities. Our approach leads to an O(min(n 2=3 ; m 1=2 )m log( n 2 m ) log U) time bound for a network with n vertices, m arcs, and integral arc capacities in the range [1; : : : ; U ]. This is a fundamental improvement over the previous time bounds. We also improve bounds for the GomoryHu tree problem, the parametric flow problem, and the approximate st cut problems. 1 Introduction The maximum flow problem and its dual, the minimum cut problem, are classical combinatorial problems with a wide variety of scientific and engineering applications. The maximum flow problem and related flow and cut problems have been studied intensively for over three decades. The network simplex method of Dantzig [11] for the transportation problem, published in 1951, solves the maximum flow problem as a natural special case. Soon ther...
Probabilistic Analysis of Network Flow Algorithms
 Mathematics of Operations Research
, 1995
"... This paper is concerned with the design and probabilistic analysis of algorithms for the maximumflow problem and capacitated transportation problems. These algorithms run in linear time and, under certain assumptions about the probability distribution of edge capacities, obtain an optimal solution w ..."
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Cited by 6 (1 self)
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This paper is concerned with the design and probabilistic analysis of algorithms for the maximumflow problem and capacitated transportation problems. These algorithms run in linear time and, under certain assumptions about the probability distribution of edge capacities, obtain an optimal solution with high probability. The design of our algorithms is based on the following general method, which we call the mimicking method, for solving problems in which some of the input data is deterministic and some is random with a known distribution: 1. Replace each random variable in the problem by its expectation; this gives a deterministic problem instance that has a special form, making it particularly easy to solve; 2. Solve the resulting deterministic problem instance; 3. Taking into account the actual values of the random variables, mimic the solution of the deterministic instance to obtain a nearoptimal solution to the original problem; 4. Finetune this suboptimal solution to obtain an o...