Matching is as easy as matrix inversion

  title={Matching is as easy as matrix inversion},
  author={Ketan Mulmuley and Umesh V. Vazirani and Vijay V. Vazirani},
We present a new algorithm for finding a maximum matching in a general graph. The special feature of our algorithm is that its only computationally non-trivial step is the inversion of a single integer matrix. Since this step can be parallelized, we get a simple parallel (RNC2) algorithm. At the heart of our algorithm lies a probabilistic lemma, the isolating lemma. We show other applications of this lemma to parallel computation and randomized reductions. 

Processor Efficient Parallel Matching

This paper presents a RNC algorithm with almost optimal work with respect to sequential algorithms, i.e., it uses O(nω) processors, where ω is the matrix multiplication exponent.

Parallel Algorithms for Perfect Matching

The perfect matching problem has a randomized NC-algorithm based on the Isolation Lemma of Mulmuley, Vazirani and Vazirani. We give an almost complete derandomization of the Isolation Lemma for

Guest Column: Parallel Algorithms for Perfect Matching

An almost complete derandomization of the Isolation Lemma for perfect matchings in bipartite graphs is given, giving a deterministic quasi-NC-algorithm for the bipartites perfect matching problem.

An Algebraic Matching Algorithm

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  • Mathematics, Computer Science
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Using linear algebra and ideas from the Gallai–Edmonds decomposition, this work describes a very simple yet efficient algorithm that replaces the indeterminates with constants without losing rank.

A New NCAlgorithm for Perfect Matching in Bipartite Cubic Graphs

A new approach to the problem of computing perfect matchings in fast deterministic parallel time is introduced, which yields a new algorithm which finds a perfect matching in bipartite cubic graphs in time O log n and O n n logn processors in the arbitrary CRCW PRAM model.

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The rst completely combinatorial algorithm for computing the Pfaaan in polynomial time is presented and it is shown that it can be computed in the complexity class GapL; this result was not known before.

The implementation and analysis of parallel algorithm for finding perfect matching in the bipartite graphs

This paper has attempted at implementing an algorithm for the perfect matching problem, which is also the central part of the algorithm for finding a maximum flow in a net, on a parallel computer with 12 processors.

A deterministic parallel algorithm for bipartite perfect matching

This article gives an almost complete derandomization of the Isolation Lemma for perfect matchings in bipartite graphs and presents three different ways of doing this construction with a common main idea.

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  • Computer Science, Mathematics
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It is shown that a perfect matching of a line graph can be computed in NC by using a technique of dividing the graph into kingdoms, equivalent to partitioning the edge set of a graph into edge disjoint paths of even length.



The Two-Processor Scheduling Problem is in Random NC

A key ingredient of the algorithm is a generalization of a theorem of Tutte which establishes a one-to-one correspondence between the bases of the Tutte matrix of a graph and the sets of matched nodes in maximum matchings in the graph.

Fast parallel matrix and GCD computations

Fast parallel matrix inversion algorithms

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  • Computer Science, Mathematics
    16th Annual Symposium on Foundations of Computer Science (sfcs 1975)
  • 1975
It will be shown in the sequel that the parallel arithmetic complexity of all these four problems is upper bounded by O(log2n) and the algorithms that establish this bound use a number of processors polynomial in n, disproves I. Munro's conjecture.

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  • Computer Science, Mathematics
    26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
  • 1985
Given a problem for which we want to find a good parallel solution, our first goal is to establish that it is in NC (or RNC); that is, to solve it with a parallel algorithm in time t = O(logk n) with

A las vegas rnc algorithm for maximum matching

The min-max formula for the size of a maximum matching is used to convert any Monte Carlo maximum matching algorithm into a Las Vegas (error-free) one and the resulting algorithm returns (with high probability) amaximum matching and a certificate proving that the matching is indeed maximum.

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This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.

How hard is it to marry at random? (On the approximation of the permanent)

Al though finding a perfect matching is easy and finding a Hamil tonian circuit is hard, counting perfect matchings and counting Hamiltonian circuits is equally hard, as hard as computing the number of solutions of any problem in NP.

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An O(v|v| c |E|) algoithm for finding maximum matching in general graphs

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  • Computer Science
    21st Annual Symposium on Foundations of Computer Science (sfcs 1980)
  • 1980
An 0(√|V|¿|E|) algorithm for finding a maximum matching in general graphs works in 'phases'.