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52
ON THE COMPLEXITY OF APPROXIMATING kSET PACKING
 COMPUTATIONAL COMPLEXITY
, 2006
"... Given a kuniform hypergraph, the Maximum kSet Packing problem is to find the maximum disjoint set of edges. We prove that this problem cannot be efficiently approximated to within a factor of Ω(k / ln k) unless P = NP. This improves the previous hardness of approximation factor of k/2 O( √ ln k) ..."
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Cited by 22 (0 self)
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Given a kuniform hypergraph, the Maximum kSet Packing problem is to find the maximum disjoint set of edges. We prove that this problem cannot be efficiently approximated to within a factor of Ω(k / ln k) unless P = NP. This improves the previous hardness of approximation factor of k/2 O( √ ln k) by Trevisan. This result extends to the problem of kDimensionalMatching.
Maximum matchings in planar graphs via Gaussian elimination
 ALGORITHMICA
, 2004
"... We present a randomized algorithm for finding maximum matchings in planar graphs in time O(n ω/2), where ω is the exponent of the best known matrix multiplication algorithm. Since ω < 2.38, this algorithm breaks through the O(n 1.5) barrier for the matching problem. This is the first result of this ..."
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Cited by 16 (2 self)
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We present a randomized algorithm for finding maximum matchings in planar graphs in time O(n ω/2), where ω is the exponent of the best known matrix multiplication algorithm. Since ω < 2.38, this algorithm breaks through the O(n 1.5) barrier for the matching problem. This is the first result of this kind for general planar graphs. We also present an algorithm for generating perfect matchings in planar graphs uniformly at random using O(n ω/2) arithmetic operations. Our algorithms are based on the Gaussian elimination approach to maximum matchings introduced in [1].
Algebraic structures and algorithms for matching and matroid problems
"... We present new algebraic approaches for several wellknown combinatorial problems, including nonbipartite matching, matroid intersection, and some of their generalizations. Our work yields new randomized algorithms that are the most efficient known. For nonbipartite matching, we obtain a simple, pu ..."
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Cited by 10 (2 self)
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We present new algebraic approaches for several wellknown combinatorial problems, including nonbipartite matching, matroid intersection, and some of their generalizations. Our work yields new randomized algorithms that are the most efficient known. For nonbipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n ω) where n is the number of vertices and ω is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time O(nr ω−1) for matroids with n elements and rank r that satisfy some natural conditions. This algorithm is based on new algebraic results characterizing the size of a maximum intersection in contracted matroids. Furthermore, the running time of this algorithm is essentially optimal.
Algebraic Algorithms for Matching and Matroid Problems
 SIAM JOURNAL ON COMPUTING
, 2009
"... We present new algebraic approaches for two wellknown combinatorial problems: nonbipartite matching and matroid intersection. Our work yields new randomized algorithms that exceed or match the efficiency of existing algorithms. For nonbipartite matching, we obtain a simple, purely algebraic algori ..."
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Cited by 10 (0 self)
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We present new algebraic approaches for two wellknown combinatorial problems: nonbipartite matching and matroid intersection. Our work yields new randomized algorithms that exceed or match the efficiency of existing algorithms. For nonbipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n ω) where n is the number of vertices and ω is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time O(nr ω−1) for matroids with n elements and rank r that satisfy some natural conditions.
Zigzag Persistent Homology in Matrix Multiplication Time
 IN: PROCEEDINGS OF THE TWENTYSEVENTH ANNUAL SYMPOSIUM ON COMPUTATIONAL GEOMETRY, 2011
, 2011
"... We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n × n matrices in M(n) time, our al ..."
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Cited by 9 (0 self)
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We present a new algorithm for computing zigzag persistent homology, an algebraic structure which encodes changes to homology groups of a simplicial complex over a sequence of simplex additions and deletions. Provided that there is an algorithm that multiplies two n × n matrices in M(n) time, our algorithm runs in O(M(n) + n 2 log 2 n) time for a sequence of n additions and deletions. In particular, the running time is O(n 2.376), by result of Coppersmith and Winograd. The fastest previously known algorithm for this problem takes O(n 3) time in the worst case.
Approximating Maximum Weight Matching in Nearlinear Time
"... Given a weighted graph, the maximum weight matching problem (MWM) is to find a set of vertexdisjoint edges with maximum weight. In the 1960s Edmonds showed that MWMs can be found in polynomial time. At present the fastest MWM algorithm, due to Gabow and Tarjan, runs in Õ(m √ n) time, where m and n ..."
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Cited by 9 (1 self)
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Given a weighted graph, the maximum weight matching problem (MWM) is to find a set of vertexdisjoint edges with maximum weight. In the 1960s Edmonds showed that MWMs can be found in polynomial time. At present the fastest MWM algorithm, due to Gabow and Tarjan, runs in Õ(m √ n) time, where m and n are the number of edges and vertices in the graph. Surprisingly, restricted versions of the problem, such as computing (1 − ɛ)approximate MWMs or finding maximum cardinality matchings, are not known to be much easier (on sparse graphs). The best algorithms for these problems also run in Õ(m √ n) time. In this paper we present the first nearlinear time algorithm for computing (1 − ɛ)approximate MWMs. Specifically, given an arbitrary realweighted graph and ɛ> 0, our algorithm computes such a matching in O(mɛ −2 log 3 n) time. The previous best approximate MWM algorithm with comparable running time could only guarantee a (2/3 − ɛ)approximate solution. In addition, we present a faster algorithm, running in O(m log n log ɛ −1) time, that computes a (3/4−ɛ)approximate MWM.
Improved Distributed Approximate Matching
"... We present improved algorithms for finding approximately optimal matchings in both weighted and unweighted graphs. For unweighted graphs, we give an algorithm providing (1 − ɛ)approximation in O(log n) time for any constant ɛ> 0. This result improves on the classical 1approximation due 2 to Israel ..."
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Cited by 9 (2 self)
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We present improved algorithms for finding approximately optimal matchings in both weighted and unweighted graphs. For unweighted graphs, we give an algorithm providing (1 − ɛ)approximation in O(log n) time for any constant ɛ> 0. This result improves on the classical 1approximation due 2 to Israeli and Itai. As a byproduct, we also provide an improved algorithm for unweighted matchings in bipartite graphs. In the context of weighted graphs, we give another algorithm which provides ( 1 − ɛ) approximation in general 2 graphs in O(log n) time. The latter result improves on the − ɛ)approximation in O(log n) time. known ( 1 4
Quantum algorithms for matching and network flow
 IN PROCEEDINGS OF THE 23RD INTERNATIONAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE
, 2006
"... We present quantum algorithms for some graph problems: finding a maximal bipartite matching in time O(n √ m log n), finding a maximal nonbipartite matching in time O(n 2 ( p m/n+log n) log n), and finding a maximal flow in an integer network in time O(min(n 7/6 √ m · U 1/3, √ nUm) log n), where n ..."
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Cited by 6 (0 self)
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We present quantum algorithms for some graph problems: finding a maximal bipartite matching in time O(n √ m log n), finding a maximal nonbipartite matching in time O(n 2 ( p m/n+log n) log n), and finding a maximal flow in an integer network in time O(min(n 7/6 √ m · U 1/3, √ nUm) log n), where n is the number of vertices, m is the number of edges, and U ≤ n 1/4 is an upper bound on the capacity of an edge.
Fast algorithms for (max,min)matrix multiplication and bottleneck shortest paths
 In Proc. 19th SODA
, 2009
"... Given a directed graph with a capacity on each edge, the allpairs bottleneck paths (APBP) problem is to determine, for all vertices s and t, the maximum flow that can be routed from s to t. For dense graphs this problem is equivalent to that of computing the (max, min)transitive closure of a realv ..."
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Cited by 6 (0 self)
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Given a directed graph with a capacity on each edge, the allpairs bottleneck paths (APBP) problem is to determine, for all vertices s and t, the maximum flow that can be routed from s to t. For dense graphs this problem is equivalent to that of computing the (max, min)transitive closure of a realvalued matrix. In this paper, we give a (max, min)matrix multiplication algorithm running in time O(n (3+ω)/2) ≤ O(n 2.688), where ω is the exponent of binary matrix multiplication. Our algorithm improves on a recent O(n 2+ω/3) ≤ O(n 2.792)time algorithm of Vassilevska, Williams, and Yuster. Although our algorithm is slower than the best APBP algorithm on vertex capacitated graphs, running in O(n 2.575) time, it is just as efficient as the best algorithm for computing the dominance product, a problem closely related to (max, min)matrix multiplication. Our techniques can be extended to give subcubic algorithms for related bottleneck problems. The allpairs bottleneck shortest paths problem (APBSP) asks for the maximum flow that can be routed along a shortest path. We give an APBSP algorithm for edgecapacitated graphs running in O(n (3+ω)/2) time and a slightly faster O(n 2.657)time algorithm for vertexcapactitated graphs. The second algorithm significantly improves on an O(n2.859)time APBSP algorithm of Shapira, Yuster, and Zwick. Our APBSP algorithms make use of new hybrid products we call the distancemaxmin product and dominancedistance product. 1