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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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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.
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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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
Algorithmic Applications of BaurStrassen’s Theorem: Shortest Cycles, Diameter and Matchings
"... Abstract—Consider a directed or undirected graph with integral edge weights in [−W, W]. This paper introduces a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the BaurStrassen Theorem and Strojohann’s determinant algorithm. For directed ..."
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Abstract—Consider a directed or undirected graph with integral edge weights in [−W, W]. This paper introduces a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the BaurStrassen Theorem and Strojohann’s determinant algorithm. For directed and undirected graphs without negative cycles we obtain simple Õ(Wnω) running time algorithms for finding a shortest cycle, computing the diameter or radius, and detecting a negative weight cycle. For each of these problems we unify and extend the class of graphs for which Õ(Wnω) time algorithms are known. In particular no such algorithms were known for any of these problems in undirected graphs with (potentially) negative weights. We also present an Õ(Wnω) time algorithm for minimum weight perfect matching. This resolves an open problem posed by Sankowski in 2006, who presented such an algorithm for bipartite graphs. Our algorithm uses a novel combinatorial interpretation of the linear program dual for minimum perfect matching. We believe this framework will find applications for finding larger spectra of related problems. As an example we give a simple Õ(Wnω) time algorithm to find all the vertices that lie on cycles of length at most t, for given t. This improves an Õ(Wn ω t) time algorithm of Yuster. Keywordsshortest cycles; diameter; radius; minimum weight perfect matchings; matrix multiplication I.
Even Factors, Jump Systems, and Discrete Convexity
, 2007
"... A jump system, which is a set of integer lattice points with an exchange property, is an extended concept of a matroid. Some combinatorial structures such as the degree sequences of the matchings in an undirected graph are known to form a jump system. On the other hand, the maximum even factor probl ..."
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A jump system, which is a set of integer lattice points with an exchange property, is an extended concept of a matroid. Some combinatorial structures such as the degree sequences of the matchings in an undirected graph are known to form a jump system. On the other hand, the maximum even factor problem is a generalization of the maximum matching problem into digraphs. When the given digraph has a certain property called oddcyclesymmetry, this problem is polynomially solvable. The main result of this paper is that the degree sequences of all even factors in a digraph form a jump system if and only if the digraph is oddcyclesymmetric. Furthermore, as a generalization, we show that the weighted even factors induce Mconvex (Mconcave) functions on jump systems. These results suggest that even factors are a natural generalization of matchings and the assumption of oddcyclesymmetry of digraphs is essential.
Strongest Postcondition of Unstructured Programs
, 2009
"... To avoid exponential explosion, program verifiers turn the program into a passive form before generating verification conditions. A little known fact is that the passive form makes it easy to use a strongest postcondition calculus to derive the verification condition. In the first part of this paper ..."
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To avoid exponential explosion, program verifiers turn the program into a passive form before generating verification conditions. A little known fact is that the passive form makes it easy to use a strongest postcondition calculus to derive the verification condition. In the first part of this paper, the passivation phase is defined precisely enough to allow a study of its algorithmic properties. In the second part, the weakest precondition and strongest postcondition methods are presented in a unified way and then compared empirically.
Dynamic Matchings in Convex Bipartite Graphs
"... Abstract. We consider the problem of maintaining a maximum matching in a convex bipartite graph G = (V, E) under a set of update operations which includes insertions and deletions of vertices and edges. It is not hard to show that it is impossible to maintain an explicit representation of a maximum ..."
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Abstract. We consider the problem of maintaining a maximum matching in a convex bipartite graph G = (V, E) under a set of update operations which includes insertions and deletions of vertices and edges. It is not hard to show that it is impossible to maintain an explicit representation of a maximum matching in sublinear time per operation, even in the amortized sense. Despite this difficulty, we develop a data structure which maintains the set of vertices that participate in a maximum matching in O(log 2 V ) amortized time per update and reports the status of a vertex (matched or unmatched) in constant worstcase time. Our structure can report the mate of a matched vertex in the maximum matching in worstcase O(min{k log 2 V +log V , V  log V }) time, where k is the number of update operations since the last query for the same pair of vertices was made. In addition, we give an O ( � V  log 2 V )time amortized bound for this pair query. 1
An Algebraic Algorithm for Weighted Linear Matroid Intersection
"... We present a new algebraic algorithm for the classical problem of weighted matroid intersection. This problem generalizes numerous wellknown problems, such as bipartite matching, network flow, etc. Our algorithm has running time Õ(nrω−1 W 1+ɛ) for linear matroids with n elements and rank r, where ω ..."
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We present a new algebraic algorithm for the classical problem of weighted matroid intersection. This problem generalizes numerous wellknown problems, such as bipartite matching, network flow, etc. Our algorithm has running time Õ(nrω−1 W 1+ɛ) for linear matroids with n elements and rank r, where ω is the matrix multiplication exponent, and W denotes the maximum weight of any element. This algorithm is the fastest known when W is small. Our approach builds on the
9 Linear Time Approximation Algorithms for Degree Constrained Subgraph Problems
"... Summary. Many realworld problems require graphs of such large size that polynomial time algorithms are too costly as soon as their runtime is superlinear. Examples include problems in VLSIdesign or problems in bioinformatics. For such problems the question arises: What is the best solution that ca ..."
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Summary. Many realworld problems require graphs of such large size that polynomial time algorithms are too costly as soon as their runtime is superlinear. Examples include problems in VLSIdesign or problems in bioinformatics. For such problems the question arises: What is the best solution that can be obtained in linear time? We survey linear time approximation algorithms for some classical problems from combinatorial optimization, e.g. matchings and branchings. For many combinatorial optimization problems arising from realworld applications, efficient, i.e., polynomial time algorithms are known for computing an optimum solution. However, there exist several applications for which the input size can easily exceed 10 9. In such cases polynomial time algorithms with a runtime that is quadratic