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84
A Combinatorial, Strongly PolynomialTime Algorithm for Minimizing Submodular Functions
, 2000
"... algorithm for minimizing submodular functions, answering an open question posed in 1981 by GrStschel, Lovsz, and Schrijver. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting ..."
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Cited by 65 (6 self)
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algorithm for minimizing submodular functions, answering an open question posed in 1981 by GrStschel, Lovsz, and Schrijver. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the largest length of the function value. The paper also presents a strongly polynomialtime version that runs in time bounded by a polynomial in the size of the underlying set independent of the function value.
A fully combinatorial algorithm for submodular function minimization
 J. COMBIN. THEORY
"... This paper presents a new simple algorithm for minimizing submodular functions. For integer valued submodular functions, the algorithm runs in O(n6EO log nM) time, where n is the cardinality of the ground set, M is the maximum absolute value of the function value, and EO is the time for function eva ..."
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Cited by 51 (8 self)
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This paper presents a new simple algorithm for minimizing submodular functions. For integer valued submodular functions, the algorithm runs in O(n6EO log nM) time, where n is the cardinality of the ground set, M is the maximum absolute value of the function value, and EO is the time for function evaluation. The algorithm can be improved to run in O((n4EO+n 5) log nM) time. The strongly polynomial version of this faster algorithm runs in O((n5EO + n6) log n) time for real valued general submodular functions. These are comparable to the best known running time bounds for submodular function minimization. The algorithm can also be implemented in strongly polynomial time using only additions, subtractions, comparisons, and the oracle calls for function evaluation. This is the first fully combinatorial submodular function minimization algorithm that does not rely on the scaling method.
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.
A General Framework for Vertex Orderings, with Applications to Netlist Clustering
, 1996
"... We present a general framework for the construction of vertex orderings for netlist clustering. Our WINDOW algorithm constructs an ordering by iteratively adding the vertex with highest attraction to the existing ordering. Variant choices for the attraction function allow our framework to subsume ma ..."
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Cited by 34 (12 self)
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We present a general framework for the construction of vertex orderings for netlist clustering. Our WINDOW algorithm constructs an ordering by iteratively adding the vertex with highest attraction to the existing ordering. Variant choices for the attraction function allow our framework to subsume many graph traversals and clustering objectives from the literature. The DPRP method of [3] is then applied to optimally split the ordering into a kway clustering. Our approach is adaptable to userspecified cluster size constraints. Experimental results for clustering and multiway partitioning are encouraging. 1 Introduction A netlist hypergraph H(V; E) consists of a set of modules (vertices) V = fv 1 ; v 2 ; : : : ; v n g and a set of nets (hyperedges) E = fe 1 ; e 2 ; : : : ; e m g. A cluster C i is a nonempty subset of V , and a kway clustering P k is a set of k clusters such that every v i 2 V belongs to exactly one cluster in P k . We study the following problem: The kWay Cl...
Computing All Small Cuts in an Undirected Network
 SIAM Journal on Discrete Mathematics
, 1994
"... : Let (N ) denote the weight of a minimum cut in an edgeweighted undirected network N , and n and m denote the numbers of vertices and edges, respectively. It is known that O(n 2k ) is an upper bound on the number of cuts with weights less than k(N ), where k 1 is a given constant. This paper rs ..."
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Cited by 32 (2 self)
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: Let (N ) denote the weight of a minimum cut in an edgeweighted undirected network N , and n and m denote the numbers of vertices and edges, respectively. It is known that O(n 2k ) is an upper bound on the number of cuts with weights less than k(N ), where k 1 is a given constant. This paper rst shows that all cuts of weights less than k(N ) can be enumerated in O(m 2 n + n 2k m) time without using the maximum ow algorithm. The paper then proves for k < 4 3 that 0 n 2 is a tight upper bound on the number of cuts of weights less than k(N ), and that all those cuts can be enumerated in O(m 2 n+mn 2 log n) time. Keywords: minimum cuts, graphs, edgesplitting, polynomial algorithm Abbreviated title: Computing Small Cuts AMS subject classications: 05C35, 05C40 1 Introduction Let N stand for an undirected network with its edges being weighted by nonnegative real numbers. Counting the number of cuts with small weights, and deriving upper and lower bounds on their...
A Faster Algorithm for Finding the Minimum Cut in a Directed Graph
 JOURNAL OF ALGORITHMS
, 1994
"... We consider the problem of finding the minimum capacity cut in a directed network G with n nodes. This problem has applications to network reliability and survivability and is useful in subroutines for other network optimization problems. One can use a maximum flow problem to find a minimum cut sepa ..."
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Cited by 32 (0 self)
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We consider the problem of finding the minimum capacity cut in a directed network G with n nodes. This problem has applications to network reliability and survivability and is useful in subroutines for other network optimization problems. One can use a maximum flow problem to find a minimum cut separating a designated source node s from a designated sink node t, and by varying the sink node one can find a minimum cut in G as a sequence of at most 2n 2 maximum flow problems. We then show how to reduce the running time of these 2n 2 maximum flow algorithms to the running time for solving a single maximum flow problem. The resulting running time is O(nm log(n 2 /m)) for finding the minimum cut in either a directed or an undirected network. © 1994 Academic Press, Inc. 1.
Bisubmodular Function Minimization
 Mathematical Programming
, 2000
"... This paper presents the rst combinatorial, polynomialtime algorithm for minimizing bisubmodular functions, extending the scaling algorithm for submodular function minimization due to Iwata, Fleischer, and Fujishige. A bisubmodular function arises as a rank function of a deltamatroid. The scali ..."
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Cited by 30 (4 self)
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This paper presents the rst combinatorial, polynomialtime algorithm for minimizing bisubmodular functions, extending the scaling algorithm for submodular function minimization due to Iwata, Fleischer, and Fujishige. A bisubmodular function arises as a rank function of a deltamatroid. The scaling algorithm naturally leads to the rst combinatorial polynomialtime algorithm for testing membership in deltamatroid polyhedra. Unlike the case of matroid polyhedra, it remains open to develop a combinatorial strongly polynomial algorithm for this problem. Division of Systems Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 5608531, Japan (fujishig@sys.es.osakau.ac.jp). Research partly carried out while at Forschungsinstut fur Diskrete Mathematik, Universitat Bonn. y Department of Mathematical Engineering and Information Physics, University of Tokyo, Tokyo 1138656, Japan (iwata@sr3.t.utokyo.ac.jp). 1 1 Introduction Let V be a nite none...
Recent developments in maximum flow algorithms
 In 6th Scandinavian Workshop on Algorithm Theory (SWAT
, 1998
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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 22 (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.