Scaling matrices and counting the perfect matchings in graphs

Dufossé, Fanny; Kaya, Kamer; Panagiotas, Ioannis; Uçar, Bora

doi:10.1016/j.dam.2020.07.016

Cited by 3 publications

(2 citation statements)

References 25 publications

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“…Gurvits and Samorodnitsky [11] and Linial et al [12] used matrix scaling and proposed deterministic approximations with exponential guarantees ( 2 n and e n , respectively). Dufosse et al also employed matrix scaling to improve the variance of the existing approximations in practice [13].…”

Section: Introductionmentioning

confidence: 99%

Parallel algorithms for computing sparse matrix permanents

Kaya¹

2019

Turk J Elec Eng & Comp Sci

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The permanent is an important characteristic of a matrix and it has been used in many applications.Unfortunately, it is a hard to compute and hard to approximate the immanant. For dense/full matrices, the fastest exact algorithm, Ryser, has O(2 n−1 n) complexity. In this work, a parallel algorithm, SkipPer, is proposed to exploit the sparsity within the input matrix as much as possible. SkipPer restructures the matrix to reduce the overall work, skips the unnecessary steps, and employs a coarse-grain, shared-memory parallelization with dynamic scheduling. The experiments show that SkipPer increases the performance of exact permanent computation up to 140× compared to the naive version for general matrices. Furthermore, thanks to the coarse-grain parallelization, 14 -15× speedup on average is obtained with τ = 16 threads over sequential SkipPer. Overall, by exploiting the sparsity and parallelism, the speedup is 2000× for some of the matrices used in the experimental setting. The proposed techniques in this paper can be used to significantly improve the performance of exact permanent computation by simply replacing Ryser with SkipPer, especially when the matrix is highly sparse.

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Section: Introductionmentioning

confidence: 99%

Parallel algorithms for computing sparse matrix permanents

Kaya¹

2019

Turk J Elec Eng & Comp Sci

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“…Sequential importance sampling (SIS) constructs a perfect matching sequentially by drawing each edge with weight proportional to the doubly stochastic scaling of the adjacency matrix. First suggested by [52], and actively improved and implemented by [7,23,53]. This seems to work well but has eluded theoretical underpinnings.…”

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confidence: 99%

Sequential importance sampling for estimating expectations over the space of perfect matchings

Alimohammadi¹,

Diaconis²,

Roghani³

et al. 2021

Preprint

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This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a (1 − )-approximation for the number of perfect matchings of a λ-dense bipartite graph, using) samples. With size n on each side and for 1 2 > λ > 0, a λ-dense bipartite graph has all degrees greater than (λ + 1 2 )n. Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements.Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback, and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes.

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The matching polynomials of hypergraphs and weighted hypergraphs

Yang

Wang

2022

Discrete Math. Algorithm. Appl.

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Let [Formula: see text] be the set of the connected [Formula: see text]-uniform linear hypergraphs with [Formula: see text] vertices, where [Formula: see text]. The matching polynomial of a hypergraph [Formula: see text] is denoted by [Formula: see text], where [Formula: see text]. Several properties on the roots of [Formula: see text] are derived. We establish different expressions for [Formula: see text], such as a higher-order differential formula and an integral formula. A new concept is also introduced for the weighted matching polynomials of weighted hypergraphs, for which several basic calculating formulas are obtained. Based on the results we obtained, some formulas for counting the number of perfect matchings of [Formula: see text] are directly derived. Finally, for the weighted matching polynomials of the weighted loose paths and the weighted loose cycles, we not only obtain their recursive formulas, but also deduce their specific expression formulas.

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Scaling matrices and counting the perfect matchings in graphs

Abstract: We investigate efficient randomized methods for approximating the number of perfect matchings in bipartite graphs and general graphs. Our approach is based on assigning probabilities to edges.

Cited by 3 publications

References 25 publications

Parallel algorithms for computing sparse matrix permanents

Parallel algorithms for computing sparse matrix permanents

Sequential importance sampling for estimating expectations over the space of perfect matchings

The matching polynomials of hypergraphs and weighted hypergraphs

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