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The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Faster ShortestPath Algorithms for Planar Graphs
 STOC 94
, 1994
"... We give a lineartime algorithm for singlesource shortest paths in planar graphs with nonnegative edgelengths. Our algorithm also yields a lineartime algorithm for maximum flow in a planar graph with the source and sink on the same face. The previous best algorithms for these problems required\O ..."
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Cited by 167 (14 self)
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We give a lineartime algorithm for singlesource shortest paths in planar graphs with nonnegative edgelengths. Our algorithm also yields a lineartime algorithm for maximum flow in a planar graph with the source and sink on the same face. The previous best algorithms for these problems required\Omega\Gamma n p log n) time where n is the number of nodes in the input graph. For the case where negative edgelengths are allowed, we give an algorithm requiring O(n 4=3 log nL) time, where L is the absolute value of the most negative length. Previous algorithms for shortest paths with negative edgelengths required \Omega\Gamma n 3=2 ) time. Our shortestpath algorithm yields an O(n 4=3 log n)time algorithm for finding a perfect matching in a planar bipartite graph. A similar improvement is obtained for maximum flow in a directed planar graph.
A Discrete Global Minimization Algorithm for Continuous Variational Problems
, 2004
"... In this paper, we apply the ideas from combinatorial optimization to find globally optimal solutions to continuous variational problems. At the heart of our method is an algorithm to solve for globally optimal discrete minimal surfaces. This discrete surface problem is a natural generalization of ..."
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Cited by 23 (0 self)
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In this paper, we apply the ideas from combinatorial optimization to find globally optimal solutions to continuous variational problems. At the heart of our method is an algorithm to solve for globally optimal discrete minimal surfaces. This discrete surface problem is a natural generalization of the planargraph shortest path problem.
A LinearProcessor PolylogTime Algorithm for Shortest Paths in Planar Graphs
, 1993
"... We give an algorithm requiring polylog time and a linear number of processors to solve singlesource shortest paths in directed planar graphs, boundedgenus graphs, and 2dimensional overlap graphs. More generally, the algorithm works for any graph provided with a decomposition tree constructed using ..."
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Cited by 17 (6 self)
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We give an algorithm requiring polylog time and a linear number of processors to solve singlesource shortest paths in directed planar graphs, boundedgenus graphs, and 2dimensional overlap graphs. More generally, the algorithm works for any graph provided with a decomposition tree constructed using sizeO( p n polylog n) separators.
Characterizing Multiterminal Flow Networks and Computing Flows in Networks of Bounded Treewidth
, 1998
"... We show that if a flow network has k input/output terminals (for the traditional maximumflow problem, k = 2), its external flow pattern (the possible values of flow into and out of the terminals) has two characterizations of size independent of the total number of vertices: a set of 2 k + 1 inequ ..."
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Cited by 5 (0 self)
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We show that if a flow network has k input/output terminals (for the traditional maximumflow problem, k = 2), its external flow pattern (the possible values of flow into and out of the terminals) has two characterizations of size independent of the total number of vertices: a set of 2 k + 1 inequalities in k variables representing flow values at the terminals, and a mimicking network with at most 2 2 k vertices and the same external flow pattern as the original network. For the case in which the underlying graph has bounded treewidth, we present sequential and parallel algorithms that can compute these characterizations as well as a flow consistent with any desired feasible external flow (including a maximum flow between two given terminals) . For constant k, the sequential algorithm runs in O(n) time on 1 A preliminary version of this paper was presented at the 6th Annual ACMSIAM Symposium on Discrete Algorithms in January 1995. 2 MaxPlanckInstitut fur Informatik, D6612...
An Efficient Parallel Algorithm for MinCost Flow on Directed SeriesParallel Networks
 Proceedings of Seventh International Parallel Processing Symposium
, 1993
"... We consider the problem of finding the minimum cost of a feasible flow in directed seriesparallel networks. We allow realvalued lower and upper bounds for the flows on edges. While strongly polynomialtime algorithms are known for this problem on arbitrary networks, it is known to be "hard" for pa ..."
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We consider the problem of finding the minimum cost of a feasible flow in directed seriesparallel networks. We allow realvalued lower and upper bounds for the flows on edges. While strongly polynomialtime algorithms are known for this problem on arbitrary networks, it is known to be "hard" for parallelization. We develop, for the first time, an efficient NC algorithm to solve the mincost flow problem on directed seriesparallel networks partially solving a problem posed by Booth and Tarjan [6, 5]. Our algorithm takes O(log 2 m) time using O(m= log m) processors on an EREW PRAM and it is optimal with respect to Booth and Tarjan's algorithm with running time O(m log m). The algorithm owes it's efficiency to the tree contraction technique and using simple data structures for flow list manipulations as opposed to finger search trees. 1 Introduction Let G = (V; E) be a directed network with two distinguished vertices s and t called the source and the sink respectively. For each e = ...
Parallel and Dynamic ShortestPath Algorithms for Sparse Graphs
, 1995
"... ere capable of anything and instilling in us a desire to be the best in whatever we did. I would also like to thank my high school teachers Mr. Jaypal Chandra and Ms. Bhuvaneshvari for showing me that education could be fun, and Professors. M.V. Tamhankar, and H. Subramanian for some truly inspiring ..."
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ere capable of anything and instilling in us a desire to be the best in whatever we did. I would also like to thank my high school teachers Mr. Jaypal Chandra and Ms. Bhuvaneshvari for showing me that education could be fun, and Professors. M.V. Tamhankar, and H. Subramanian for some truly inspiring courses in mathematics. At Brown, I would like to thank Professors Philip Klein, Roberto Tamassia, and Jeff Vitter for advising this thesis and for teaching me much of what I know. I would like to thank Prof. Vitter for introducing me to research and for his confidence in my abilities. His constant encouragement kept me motivated during times when the going was tough. I would like to thank Prof. Tamassia for encouraging my interest in dynamic graph algorithms and for suggesting the problem solved in Chapter 5. A large portion of the results in this thesis were obtained in joint work with Prof. Phil Klein. I would like to thank him for his boundless enthusiasm for research and for the innume
Using Cycles and Scaling in Parallel Algorithms
, 1989
"... We introduce the technique of decomposing an undirected graph by finding a maximal set of edgedisjoint cycles. We give a parallel algorithm to find this decomposition in O(log n) time on (m + n)= log n processors. We then use this decomposition to give the first efficient parallel algorithm for fin ..."
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We introduce the technique of decomposing an undirected graph by finding a maximal set of edgedisjoint cycles. We give a parallel algorithm to find this decomposition in O(log n) time on (m + n)= log n processors. We then use this decomposition to give the first efficient parallel algorithm for finding an approximation to a minimum cycle cover. Our algorithm finds a cycle cover whose size is within a factor of O(1 + n log n m+n ) of the minimum sized cover using O(log 2 n) time on (m + n)= log n processors. We also generalize these algorithms to weighted graphs with running times that are a factor of O(log C) slower than their unweighted counterparts, where C is the largest weight in the graph. Finally, we show how to use scaling to develop parallel algorithms for the assignment problem in which the number of processors used is independent of the magnitude of the edge costs. This leads to algorithms for the assignment problem that do less work than any known RNC algorithms for th...
PARALLEL NESTED DISSECTION i 3R PATH ALGEBRA COMPUTATIONS
, 1986
"... Th/s paper extends the authors " parallel nested dissection algorithm of [13] originally devised for solving sparse lhteaf systems. We present a class of new applications of the nested dissection method, this time to path algebra compulal~,.hs (in both car~ of single source and all pair paths), wher ..."
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Th/s paper extends the authors " parallel nested dissection algorithm of [13] originally devised for solving sparse lhteaf systems. We present a class of new applications of the nested dissection method, this time to path algebra compulal~,.hs (in both car~ of single source and all pair paths), where the, ~ ~ problem is defined by a symmetric malrix.4 who ~ asem¢is4~i graph G with n verticex is planar. We substamlially improce the known algorithms for path algebra problems of that l~mnml class; this has further applications to maxinmm flow and minimum cut woblem ~ in an undirected planar network and to the feasibility testing of a multicommodity flow hi a planar networlL graph computations * path algebras * parallel algorithms • network flow I. Intro&actkm In this ~*,per we substantially improve the known par~.el algorithms for several problems of practical ~aterest which can be reduced to path algebra computations. Gondran and Min~ux [4, pp. 4142, 7581] list the applications of path algebras to the problems of: veh/cle routing,, investment and stock control, dynamic progrmnming with discrete states and discrete time, network optiniization, artificial intelligence and pattern refognition, labyrinths and mathematical <~games, enc~di ~ and decoding of information: ~ompare also Lawler [8], Tarjan [17,18]. [4, pp. 84102] thoroughly investigates general algorithms for such problems based on matrix operatio~ in d/o/ds, see next sections. We propose a substantial improvement of Ihese. gco ~ algorithms in the important case wherethe input matrix A is associated with an uadirected planar graph or, more generally, wi ~ a graph from the