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27
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 (3 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.
Practical Problem Solving with Cutting Plane Algorithms in Combinatorial Optimization
, 1994
"... Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many p ..."
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Cited by 23 (5 self)
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Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.
Distance approximation in boundeddegree and general sparse graphs
 In Proceedings of the Tenth International Workshop on Randomization and Computation (RANDOM
, 2006
"... We address the problem of approximating the distance of bounded degree and general sparse graphs from having some predetermined graph property P. Namely, we are interested in sublinear algorithms for estimating the fraction of edges that should be added to / removed from a graph so that it obtains P ..."
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Cited by 15 (5 self)
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We address the problem of approximating the distance of bounded degree and general sparse graphs from having some predetermined graph property P. Namely, we are interested in sublinear algorithms for estimating the fraction of edges that should be added to / removed from a graph so that it obtains P. This fraction is taken with respect to a given upper bound m on the number of edges. In particular, for graphs with degree bound d over n vertices, m = dn. To perform such an approximation the algorithm may ask for the degree of any vertex of its choice, and may ask for the neighbors of any vertex. The problem of estimating the distance to having a property was first explicitly addressed by Parnas et. al. (ECCC 2004). In the context of graphs this problem was studied by Fischer and Newman (FOCS 2005) in the densegraphs model. In this model the fraction of edge modifications is taken with respect to n 2, and the algorithm may ask for the existence of an edge between any pair of vertices of its choice. Fischer and Newman showed that every graph property that has a testing algorithm in this model with query complexity that is independent of the size of the graph, also has a distanceapproximation algorithm with query complexity that is independent of the size of the graph. In this work we focus on boundeddegree and general sparse graphs, and give algorithms for all properties that were shown to have efficient testing algorithms by Goldreich and Ron (Algorithmica, 2002). Specifically, these properties are kedge connectivity, subgraphfreeness (for constant size subgraphs), being a Eulerian graph, and cyclefreeness. A variant of our subgraphfreeness algorithm approximates the size of a minimum vertex cover of a graph in sublinear time. This approximation improves on a recent result of Parnas and Ron (ECCC 2005).
Label Selection on Graphs
"... We investigate methods for selecting sets of labeled vertices for use in predicting the labels of vertices on a graph. We specifically study methods which choose a single batch of labeled vertices (i.e. offline, non sequential methods). In this setting, we find common graph smoothness assumptions di ..."
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Cited by 13 (1 self)
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We investigate methods for selecting sets of labeled vertices for use in predicting the labels of vertices on a graph. We specifically study methods which choose a single batch of labeled vertices (i.e. offline, non sequential methods). In this setting, we find common graph smoothness assumptions directly motivate simple label selection methods with interesting theoretical guarantees. These methods bound prediction error in terms of the smoothness of the true labels with respect to the graph. Some of these bounds give new motivations for previously proposed algorithms, and some suggest new algorithms which we evaluate. We show improved performance over baseline methods on several real world data sets. 1
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...
A branch and cut approach to the cardinality constrained circuit problem
 MATH. PROGRAM., SER. A 91: 307–348 (2002)
, 2002
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Efficient Algorithms for Computing All Low st Edge Connectivities and Related
 Problems, Proc. of the 18th Annual ACMSIAM Symposium on Discrete Algorithms
, 2007
"... Given an undirected unweighted graph G =(V,E) andan integer k ≥ 1, we consider the problem of computing the edge connectivities of all those (s, t) vertex pairs, whose edge connectivity is at most k. We present an algorithm with expected running time Õ(m + nk3) for this problem, where V  = n and ..."
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Cited by 8 (0 self)
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Given an undirected unweighted graph G =(V,E) andan integer k ≥ 1, we consider the problem of computing the edge connectivities of all those (s, t) vertex pairs, whose edge connectivity is at most k. We present an algorithm with expected running time Õ(m + nk3) for this problem, where V  = n and E  = m. Our output is a weighted tree T whose nodes are the sets V1,V2,...,Vℓ of a partition of V, with the property that the edge connectivity in G between any two vertices s ∈ Vi and t ∈ Vj, fori� = j, is equal to the weight of the lightest edge on the path between Vi and Vj in T. Also, two vertices s and t belong to the same Vi for any i if and only if they have an edge connectivity greater than k. Currently, the best algorithm for this problem needs to compute allpairs mincuts in an O(nk) edge graph; this
Designing multicommodity flow trees
 Inform. Process. Lett
, 1994
"... The traditional multicommodity flow problem assumes a given flow network in which multiple commodities are to be maximally routed in response to given demands. This paper considers the multicommodity flow networkdesigu problem: given a set of multicommodity flow demands, find a network subject t ..."
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Cited by 6 (0 self)
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The traditional multicommodity flow problem assumes a given flow network in which multiple commodities are to be maximally routed in response to given demands. This paper considers the multicommodity flow networkdesigu problem: given a set of multicommodity flow demands, find a network subject to certain constraints such that the commodities can be maximally routed. This paper focuses on the case when the network is required to be a tree. The main result is an approximation algorithm for the case when the tree is required to be of constant degree. The algorithm reduces the problem to the minimumweight balancedseparator problem; the performance guarantee of the algorithm is within a factor of 4 of the performance guarantee of the balanced~parator procedure. If Leighton and P~o's balancedseparator proced'~e is used, the performance guarantee is O(logn). 1
Dynamic Graph Clustering Using MinimumCut Trees
"... Abstract. Algorithms or target functions for graph clustering rarely admit quality guarantees or optimal results in general. Based on properties of minimumcut trees, a clustering algorithm by Flake et al. does however yield such a provable guarantee. We show that the structure of minimumstcuts i ..."
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Cited by 5 (1 self)
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Abstract. Algorithms or target functions for graph clustering rarely admit quality guarantees or optimal results in general. Based on properties of minimumcut trees, a clustering algorithm by Flake et al. does however yield such a provable guarantee. We show that the structure of minimumstcuts in a graph allows for an efficient dynamic update of minimumcut trees, and present a dynamic graph clustering algorithm that maintains a clustering fulfilling this quality quarantee, and that effectively avoids changing the clustering. Experiments on realworld dynamic graphs complement our theoretical results. 1
Practical Performance of Efficient Minimum Cut Algorithms
, 1997
"... In the early nineties, three major exciting new developments (and some ramifications) in the computation of minimum capacity cuts occurred and these developments motivated us to evaluate the old and new methods experimentally. We provide a brief overview of the most important algorithms for the mini ..."
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Cited by 4 (1 self)
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In the early nineties, three major exciting new developments (and some ramifications) in the computation of minimum capacity cuts occurred and these developments motivated us to evaluate the old and new methods experimentally. We provide a brief overview of the most important algorithms for the minimum cut problem and compare these methods both on problem instances from the literature and graphs originating from the solution of the traveling salesman problem by branchandcut. 1. Introduction Computer programs that compute minimum capacity cuts in undirected graphs with nonnegative edge capacities belong to the most intensively used basic tools in optimization. Besides the obvious direct application of deciding the degree of connectivity of a given network, the main reason is that various separation routines in cutting plane algorithms depend on the practically efficient computation of minimum capacity cuts. The most prominent example is the separation of subtour elimination constrain...