K. Mulmuley, U.V. Vazirani, and V.V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105--114, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....This algorithm simplifies to an RNC algorithm for deciding the existence of a spanning tree in an unweighted 3 uniform hypergraph and implies a fully polynomial approximation scheme if the edgeweights are given in binary. To achieve this result, we combine ideas from Mulmuley, Vazirani, Vazirani [18] which they used to obtain an RNC algorithm for finding a perfect matching in a given graph with ideas from Lov asz [17] resp. Camerini, Galbiati and Maffioli [5] where they presented a random pseudo polynomial time algorithm for the general problem of finding a base of specified value in a ....

....time algorithm for the weighted case . In this section we develop a randomized parallel algorithm for the minimum spanning tree problem in 3 uniform hypergraphs. It is also based on the algebraic approach of Lov asz [17] The parallelization is based on ideas from Mulmuley, Vazirani, Vazirani [18]. We start with some definitions. Let A = a ij ) be a skew symmetric matrix (i.e. A = GammaA) of size 2n Theta 2n and let P be the set of all partitions of f1; 2ng into two element sets. For an element p = ffi 1 ; i 2 g; fi 2n Gamma1 ; i 2n gg of P we denote by oe(p) the ....

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K. Mulmuley, U. Vazirani, and V. Vazirani, Matching is as easy as matrix inversion, Combinatorica 7 (1987), 105--113.

....or all unbounded fan in circuits the analogous inclusion of the Boolean in the arithmetic class is trivial. In this paper we prove both inclusions, namely NL=poly PhiL=poly and . We observe that the right technique to use is applying the Isolation Lemma of Mulmuley Vazirani Vazirani [MVV]. We note that [MVV] showed how the Isolation Lemma can be used to rederive the Valiant Vazirani result for the clique function. The Isolation Lemma can be embedded in graph reachability, generating from an arbitrary graph which has an st path another graph in which such a path is unique. From ....

....fan in circuits the analogous inclusion of the Boolean in the arithmetic class is trivial. In this paper we prove both inclusions, namely NL=poly PhiL=poly and . We observe that the right technique to use is applying the Isolation Lemma of Mulmuley Vazirani Vazirani [MVV] We note that [MVV] showed how the Isolation Lemma can be used to rederive the Valiant Vazirani result for the clique function. The Isolation Lemma can be embedded in graph reachability, generating from an arbitrary graph which has an st path another graph in which such a path is unique. From this the NL=poly ....

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K. Mulmuley, U. Vazirani and V. Vazirani, "Matching is as easy as matrix inversion," In Proc. of the 19th STOC, (1987), pp. 345-354.

....by assigning higher weight to nonexisting edges. To date it is still not known whether this problem is in the class NC (i.e. admits a PRAM algorithm that runs in polylogarithmic time using polynomially many processors) Karp, Upfal and Wigderson [KUW86] showed that the problem is in RNC (see [MVV87] for a simpler and more ecient algorithm) They also showed that nding a perfect matching with maximum weight in an edge weighted graph is in RNC if the weights are bounded by a polynomial of the number of vertices in the graph. Approximating the size of the maximum matching in parallel turns ....

K. Mulmuley, U. Vazirani and V.V. Vazirani. Matching is as easy as matrix inversion. Combinatorica 7(1):105-113, 1987.

....in step 4 by taking r examples (see the algorithm) and constructs the set B. This can be done in NC . Now each active processor that corresponds to b 2 A will run r processors and define B b . The processors corresponding to b will build the graph G f (B) and find a maximal matching. In [MVV87] it is shown that maximal matching in graphs can be done in RNC . We also showed in section 4 that the number of matching we need is log s and therefore step 7 can be done efficiently in parallel. In steps 7.3, 7.4 and 8 each processor can build the hypothesis in parallel and output h. In ....

K. Mulmuley, U. V. Vazirani, V. V. Vazirani. Matching is as Easy as Matrix Inversion, ACM Symposium on Theory of Computing (STOC), 1987, 345--354.

....applications of f(OE) for some satisfiable OE then with extremely high probability one of these outputs will have a unique assignment. Theorem 1.1 follows directly from Lemma 3.1. Valiant and Vazirani s construction creates random subspaces of the assignments. Mulmuley, Vazirani and Vazirani [MVV87] give an alternate proof looking at maximal weighted cliques after putting random weights on the edges. Buhrman and Fortnow [BF97] show how Lemma 3.1 follows from earlier work by Sipser [Sip83] on Kolmogorov complexity. Gupta [Gup97] gives a construction for Lemma 3.1 that improves the ....

K. Mulmuley, U. Vazirani, and V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105--113, 1987.

....in polynomially many trials. The function mapping x to y then belongs to the class BPPSV defined by Grollmann and Selman [9] Having a monic refinement is different from the notion of probabilistically isolating a unique element in Chari, Rohatgi, and Srinivasan [4] They use the method from [19] of assigning random weights to edges so that with high probability, there is a unique minimum weight perfect matching. However, different random weightings can yield different matchings. Now we reconsider the problems of Section 1 when the generator is feasible, and when Alice and Bob share the ....

K. Mulmuley, U. Vazirani, and V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7:105--113, 1987.

.... algorithms used in approximation algorithm design (see [MNR95] for a list) the 3 4 approximation algorithm for MAX SAT due to Goemans and Williamson [GW93] a randomized parallel set cover approximation algorithm due to Rajagopalan and Vazirani [RV98] special cases of the isolation lemma of [MVV87] where the number of sets in the family is only polynomial, and several computational geometry problems (see [MRS97] for a detailed list) There is still a large class of derandomizations (some known to be achievable, others desirable) that don t seem to be achievable via our approach. We ....

....several computational geometry problems (see [MRS97] for a detailed list) There is still a large class of derandomizations (some known to be achievable, others desirable) that don t seem to be achievable via our approach. We highlight the Lov asz local lemma (see [AS92] the isolation lemma of [MVV87] and the randomized primality testing algorithms (see [MR95b] Indeed, the relevant statistical tests don t seem to be logspace computable. Other cases include highly efficient randomized algorithms, such as Karger s min cut algorithm [Kar93] it is possible to derandomize this algorithm using ....

K. Mulmuley, U. Vazirani, and V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7:105--113, 1987.

....of graphs which we cite later. The class of problems which can be solved in poly logarithmic expected time using a polynomial number of processors is called RNC (see, for example, chapter 3 of [11] and section 2.5. 5 of [22] Many matching problems can be solved in parallel using randomness ([20, 24]) and so belong to RNC. It has been stated [22] that whether a modern definition of a tractable problem in parallel computation is one can that can be solved rapidly with randomisation or one that can be solved rapidly without randomisation may ultimately depend upon whether fast parallel ....

....Karp, Upfal and Wigderson [20] described an RNC algorithm for the minimum weight perfect matching problem, which runs in O(log n log 2 (Wn) time (after the improvements of [8] using O(Wn 3:5 ) processors, where W is the maximum weight of any edge. A faster RNC algorithm was obtained in 19 [24] which runs in O(log 2 n) time using O(mWn 3:5 ) processors, where m is the number of edges. These algorithms are in RNC only if W is relatively small (that is, W = n k , for some constant k. Although finding an NC algorithm for minimum weight perfect matching seems to be hard, there are ....

K. Mulmuley, U. Vazirani, and V. Vazirani. Matching is as easy as matrix inversion. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pages 139--148, 1987.

....In this paper, we present a randomised NC (RNC) algorithm for UCFGBEQ. Theorem 1. UCFGBEQ is in RNC4 in terms of the product of n, given in unary notation and the grammar size. In general randomised NC algorithms are relatively uncommon.However, the maximum matching algorithm presented in [12] is an example of a randomised NC (RNC2 in fact) algorithm. 1.2 Outline of the paper Section 2 contains preliminaries from context free language theory and a few facts about related formal series. Section 3 presents the randomisation result, which can be viewed as a kind of sieve. The ....

K. Mulmuley, U. Vazirani, and V. Vazirani. Matching is as easy as matrix inversion. In 19th Symp. on Theory of Computing (STOC), pages 345--354. ACM, 1987.

....some cases in which these coin flips can be replaced by deterministic techniques. Randomness can improve the bounds of an algorithm in several ways. First, a randomized algorithm may solve in polylogarithmic time a problem for which there is no known NC algorithm. This is the case for matching [36, 23, 53] and for the construction of depth first search trees [1] Karp et al. 37] describe a problem in a model of parallel computation with oracles which no deterministic PRAM can solve in NC, but which can be solved using a probabilistic algorithm. Second, a randomized parallel algorithm may be more ....

....matrix, and then computing a perfect matching on a graph derived from the original graph; the resulting algorithm takes the same bounds as perfect matching. The minimum weight perfect matching problem can be solved using a factor of O(nw) more processors than needed to find a perfect matching [22, 53]; here w is the largest weight on any edge. Finding a maximal matching seems to be much easier. Israeli and Shiloach [33] give an algorithm which takes O(log 3 m) time with O(m n) CRCW processors. Israeli and Itai [32] give a randomized algorithm using the same number of processors, but ....

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K. Mulmuley, U.V. Vazirani, and V.V. Vazirani, Matching is as Easy as Matrix Inversion. 19th Symp. Theory of Computing, 1987, 345--354, and Combinatorica 7, 1, 1987, 105--114.

....that contain all the vertices, and such that no two of these edges share a vertex. This problem has been intensively studied, but like GI, it has resisted all classification attempts in terms of completeness in a class. The problem has polynomial time algorithms, and it is known to be in random NC [18, 27]. In [5] it has been proved that for any k 2, the perfect matching problem can be randomly reducible to a set in Mod k L. Together with Theorem 4.3 this implies: Corollary 5.8 Matching is reducible to GI under logarithmic space randomized reductions. Since the reduction works correctly with ....

K. Mulmuley, U. Vazirani and V. Vazirani. Matching is as easy as matrix inversion. Combinatorica 7:105--113, 1987.

....well known problems in algorithms and complexity reduce to the polynomial identity testing problem: given a multivariate polynomial p(x 1 ; x n ) over a field F , determine if the polynomial is identically zero. Algorithms for testing primality [AB99] or if a graph has a perfect matching [MVV87] for example, involve testing if a particular polynomial is equal to zero. In addition, fundamental structural results in complexity theory such as IP=PSPACE [Sha92] or the PCP theorem [AS98, ALM 98] make heavy use of identity testing. The complexity of the problem is highly representation ....

....be distinct vectors with entries in f0; 1; 2; g, and let p be a prime greater than t and . Then Pr 1 k t hD d (j) j a (k) E are distinct for all 1 j m i 1 m 2 n=t: We will use a slight improvement of an isolation lemma from [CRS95] which is based on a lemma from [MVV87] Our proof closely follows those of [CRS95, MVV87] Lemma 4 Let C be any collection of distinct linear forms in variables z 1 ; z with coefficients in the range f0; Kg. If z 1 ; z are independently chosen uniformly at random in f0; K = g, then, with ....

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K. Mulmuley, U. V. Vazirani, and V. V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7:105--113, 1987.

....the efficiency of RNC matching algorithms when the size of the maximum matching is substantially smaller than the number of vertices. We first run the above algorithm as a preprocessing step to find the vertex set B. Suppose B is of size jBj = R. Then we run the algorithm of [16] which improves [30] ) to find a perfect matching on the subgraph induced by B in parallel time O(T (R) log R) using RM(R) processors. Corollary 4.1. Let G be an n vertex m edge graph, and let R denote the size of a maximum matching in G. Then a randomized PRAM (with integer arithmetic operations and O(m log n n ....

K.Mulmuley, U.V.Vazirani and V.V.Vazirani, Matching is as easy as matrix inversion, Combinatorica 7 (1987), 105--113.

....class of languages for which there exists a uniform family of randomized Boolean circuits Cn of polynomial size and depth O(log k n) that decide the language with one sided error. For example, the problem of deciding whether a graph has a perfect matching or not can be decided in RNC 2 [36]. 2.5 Non determinism A non deterministic Turing machine does not represent a physically realizable model of computation. In this model, the machine has the ability to make guesses about possible choices in the computation. Even though the model is physically unrealizable, it is so useful in ....

Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7:105--113, 1987.

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Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7:105--113, 1987.

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K. Mulmuley, U.V. Vazirani, and V.V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105--114, 1987.

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K. Mulmuley, U. Vazirani, and V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105-113, 1987.

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Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7:105-113, 1987.

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K. Mulmuley, U. V. Vazirani, and V. V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105-113, 1987.

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K. Mulmuley, U. Vazirani, and V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7:105--113, 1987.

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U.Vazirani K.Mulmuley and V.Vazirani. Matching is as easy as Matrix inversion. STOC, pages 345-354, 1987.

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Mulmuley, K., U.V. Vazirani and V.V. Vazirani, Matching is as easy as matrix inversion. Combinatorica 7, 1987, 105-113.

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K. Mulmuley, U. Vazirani and V. Vazirani, Matching is as easy as matrix inversion. Combinatorica ?, 105-114 (1987).

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Mulmuley, K., Vazirani, U.V., Vazirani, V.V. (1987). Matching is as Easy as Matrix Inversion. Combinatorica 7(1):105-114. 9

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K. Mulmuley, U. V. Vazirani and V. B. Vazirani, Matching is as easy as matrix inversion, Combinatorica 7 (1987) 105-113.

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