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33
On the Hardness of Graph Isomorphism
 SIAM J. COMPUT
"... We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 manyone reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the stro ..."
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Cited by 32 (1 self)
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We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 manyone reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the strongest known hardness results for the graph isomorphism problem and imply a randomized logarithmic space reduction from the perfect matching problem to graph isomorphism. We also investigate hardness results for the graph automorphism problem.
A Simple Approximation Algorithm for the Weighted Matching Problem
 Information Processing Letters
, 2003
"... We present a linear time approximation algorithm with a performance ratio of 1/2 for nding a maximum weight matching in an arbitrary graph. Such a result is already known and is due to Preis [7]. ..."
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Cited by 28 (3 self)
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We present a linear time approximation algorithm with a performance ratio of 1/2 for nding a maximum weight matching in an arbitrary graph. Such a result is already known and is due to Preis [7].
Selected Topics on Assignment Problems
, 1999
"... We survey recent developments in the fields of bipartite matchings, linear sum assignment and bottleneck assignment problems and applications, multidimensional assignment problems, quadratic assignment problems, in particular lower bounds, special cases and asymptotic results, biquadratic and co ..."
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Cited by 22 (1 self)
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We survey recent developments in the fields of bipartite matchings, linear sum assignment and bottleneck assignment problems and applications, multidimensional assignment problems, quadratic assignment problems, in particular lower bounds, special cases and asymptotic results, biquadratic and communication assignment problems.
Minimum Cuts and Shortest Homologous Cycles
 SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
"... We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the spec ..."
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Cited by 20 (7 self)
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We describe the first algorithms to compute minimum cuts in surfaceembedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimumcut algorithm computes a minimumcost subgraph in every Z2homology class. We also prove that finding a minimumcost subgraph homologous to a single input cycle is NPhard.
Random PseudoPolynomial Algorithms for Exact Matroid Problems
, 1992
"... In this work we present a random pseudopolynomial algorithm for the problem of finding a base of specified value in a weighted represented matroid, subject to parity conditions. We also describe a specialized version of the algorithm suitable for finding a base of specified value in the intersectio ..."
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Cited by 16 (0 self)
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In this work we present a random pseudopolynomial algorithm for the problem of finding a base of specified value in a weighted represented matroid, subject to parity conditions. We also describe a specialized version of the algorithm suitable for finding a base of specified value in the intersection of two matroids. This result generalizes an existing pseudopolynomial algorithm for computing exact arborescences in weighted graphs. Another (simpler) specialized version of our algorithms is also presented for computing perfect matchings of specified value in weighted graphs.
RNCApproximation Algorithms for the Steiner Problem
 Proc. STACS'97
, 1997
"... . In this paper we present an RNCalgorithm for finding a minimum spanning tree in a weighted 3uniform hypergraph, assuming the edge weights are given in unary, and a fully polynomial time randomized approximation scheme if the edge weights are given in binary. From this result we then derive RNCa ..."
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Cited by 15 (0 self)
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. In this paper we present an RNCalgorithm for finding a minimum spanning tree in a weighted 3uniform hypergraph, assuming the edge weights are given in unary, and a fully polynomial time randomized approximation scheme if the edge weights are given in binary. From this result we then derive RNCapproximation algorithms for the Steiner problem in networks with approximation ratio (1 + ffl) 5=3 for all ffl ? 0. 1 Introduction In recent years, the Steiner tree problem in graphs attracted considerable attention, as well from the theoretical point of view as from its applicability, e.g., in VLSIlayout. It is rather easy to see and has been known for a long time that a minimum Steiner tree spanning a given set of terminals in a graph or network can be approximated in polynomial time up to a factor of 2, cf. e.g. Choukhmane [6] or Kou, Markowsky, Berman [14]. After a long period without any progress Zelikovsky [23], Berman and Ramaiyer [2], Zelikovsky [24], and Karpinski and Zelikovsky ...
Lower bounds in a parallel model without bit operations
 TO APPEAR IN THE SIAM JOURNAL ON COMPUTING
, 1997
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Combinatorial Approximation Algorithms for the Maximum Directed Cut Problem
 Proc. of 12th SODA
, 2001
"... We describe several combinatorial algorithms for the maximum directed cut problem. Among our results is a simple linear time 9/20approximation algorithm for the problem, and a somewhat slower 1/2approximation algorithm that uses a bipartite matching routine. No better combinatorial approximation a ..."
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Cited by 12 (0 self)
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We describe several combinatorial algorithms for the maximum directed cut problem. Among our results is a simple linear time 9/20approximation algorithm for the problem, and a somewhat slower 1/2approximation algorithm that uses a bipartite matching routine. No better combinatorial approximation algorithms are known even for the easier maximum cut problem for undirected graphs. Our algorithms do not use linear programming, nor semidefinite programming. They are based on the observation that the maximum directed cut problem is equivalent to the problem of finding a maximum independent set in the line graph of the input graph, and that the linear programming relaxation of the problem is equivalent to the problem of finding a maximum fractional independent set of that line graph. The maximum fractional independent set problem can be easily reduced to a bipartite matching problem. As a consequence of this relation, we also get that the maximum directed cut problem for bipartite digraphs can be solved in polynomial time.
Directed st Numberings, Rubber Bands, and Testing Digraph kVertex Connectivity
"... Let G = (V, E) be a directed graph and n denote V. We show that G is kvertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k I)dimensional space Rkl, ~ : V ~Rkl, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f( ..."
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Cited by 10 (2 self)
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Let G = (V, E) be a directed graph and n denote V. We show that G is kvertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k I)dimensional space Rkl, ~ : V ~Rkl, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f(v) is in the convex hull of {~(w) I (v, W) G E}. This result generalizes to directed graphs the notion of convex embedding of undirected graphs introduced by Linial, LOV6SZ and Wigderson in ‘Rubber bands, convex embedding and graph connectivity, ” Combinatorics 8 (1988), 91102. Using this characterization, a directed graph can be tested for kvertex connectivity by a Monte Carlo algorithm in time O((M(n) + nkf(k)). (log n)) with error probability < l/n, and by a Las Vegas algorithm in expected time O((lf(n)+nM(k)).k), where M(n) denotes the number of arithmetic steps for multiplying two n x n matrices (Al(n) = 0(n2.3755)). Our Monte Carlo algorithm improves on the best previous deterministic and randomized time complexities for k> no. *9; e.g., for k = @, the factor of improvement is> n0.G2. Both algorithms have processor efficient parallel versions that run in O((log n)2) time on the EREW PRAM model of computation, using a number of processors equal to (logn) times the respective sequential time complexities. Our Monte Carlo parallel algorithm improves on the number of processors used by the best previous (Monte Carlo) parallel algorithm by a factor of at least (n2/(log n)3) while having the same running time. Generalizing the notion of st numberings, we give a combinatorial construction of a directed st nulmberiug for any 2vertex connected directed graph.