A Permanent Algorithm with exp[Ω (n ^{1/3}/2 ln n )] Expected Speedup for 0-1 Matrices

Bax, Eric; Franklin, Joel

doi:10.1007/s00453-001-0072-0

Cited by 18 publications

(18 citation statements)

References 8 publications

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“…The main differences between our work and [1] are: (1) We show that for sparse matrices, an appropriate augmented matrix can be chosen deterministically (i.e. we show how to efficiently derandomize a random augmentation of the matrix); (2) We show that our algorithm works for any (real or complex valued) sparse matrix, whereas their algorithm works for any 0/1 matrix; (3) We show that our algorithm requires only O(n) space whereas their algorithm requires 2 n/2 space.…”

Section: Relation To Previous Workmentioning

confidence: 63%

“…Even this problem must be hard; it is easily seen that computing the permanent of a 0/1 matrix with at most Cn ones is also #P complete. 1 We are not aware of previous work on algorithms that specifically target this problem. As our main result we give a deterministic algorithm B to compute the permanent of any sparse n × n matrix A, and prove the following theorem: Theorem 1.1 For any constant C > 0, there is a constant > 0 such that algorithm B runs in at most (2 − ) n time on any n × n matrix A with at most Cn nonzero entries.…”

Section: This Workmentioning

confidence: 99%

“…Bax and Franklin use a finite difference formula for the permanent (this is a generalization of Ryser's formula; like Ryser's formula, it expresses 1 If we have an algorithm to compute the permanent of such a matrix that runs in time f (n) for some function f , then we have an algorithm that computes the permanent for arbritrary 0/1 matrices that runs in time f (n 2 ); the algorithm simply takes the arbitrary n×n matrix and embeds it in a diagonal n 2 ×n 2 matrix with the original matrix in the upper left corner instead of the ones. Since this matrix has the same permanent as the original matrix and at most 2n 2 − n ones, the original algorithm can be run on it to obtain the permanent in time f (n 2 ).…”

Section: Relation To Previous Workmentioning

confidence: 99%

“…Given an n × n matrix A and an n-dimensional column vector b, as in [1] we let A b be the (n + 1) × (n + 1) matrix…”

Section: Using Ryser's Formulamentioning

confidence: 99%

“…algorithm) for arbitrary matrices. More recently in [1], Bax & Franklin gave a randomized algorithm for computing the permanent of 0/1 matrices that runs in expected time exp −Ω n 1/3 2 ln n 2 n .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Computing sparse permanents faster

Servedio

Wan

2005

Information Processing Letters

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Bax and Franklin (2002) gave a randomized algorithm for exactly computing the permanent of any n × n zero-one matrix in expected time exp −Ω n 1/3 2 ln n 2 n . Building on their work, we show that for any constant C > 0 there is a constant > 0 such that the permanent of any n × n (real or complex) matrix with at most Cn nonzero entries can be computed in deterministic time (2 − ) n and space O(n). This improves on the Ω(2 n ) runtime of Ryser's algorithm for computing the permanent of an arbitrary real or complex matrix.

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Section: Relation To Previous Workmentioning

confidence: 63%

Section: This Workmentioning

confidence: 99%

Section: Relation To Previous Workmentioning

confidence: 99%

“…Given an n × n matrix A and an n-dimensional column vector b, as in [1] we let A b be the (n + 1) × (n + 1) matrix…”

Section: Using Ryser's Formulamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Computing sparse permanents faster

Servedio

Wan

2005

Information Processing Letters

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On Symbolic Approaches for Computing the Matrix Permanent

Chakraborty

Shrotri

Vardi

2019

Lecture Notes in Computer Science

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Counting the number of perfect matchings in bipartite graphs, or equivalently computing the permanent of 0-1 matrices, is an important combinatorial problem that has been extensively studied by theoreticians and practitioners alike. The permanent is #P-Complete; hence it is unlikely that a polynomial-time algorithm exists for the problem. Researchers have therefore focused on finding tractable subclasses of matrices for permanent computation. One such subclass that has received much attention is that of sparse matrices i.e. matrices with few entries set to 1, the rest being 0. For this subclass, improved theoretical upper bounds and practically efficient algorithms have been developed. In this paper, we ask whether it is possible to go beyond sparse matrices in our quest for developing scalable techniques for the permanent, and answer this question affirmatively. Our key insight is to represent permanent computation symbolically using Algebraic Decision Diagrams (ADDs). ADD-based techniques naturally use dynamic programming, and hence avoid redundant computation through memoization. This permits exploiting the hidden structure in a large class of matrices that have so far remained beyond the reach of permanent computation techniques. The availability of sophisticated libraries implementing ADDs also makes the task of engineering practical solutions relatively straightforward. While a complete characterization of matrices admitting a compact ADD representation remains open, we provide strong experimental evidence of the effectiveness of our approach for computing the permanent, not just for sparse matrices, but also for dense matrices and for matrices with "similar" rows.

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Families with Infants: A General Approach to Solve Hard Partition Problems

Golovnev

Mihajlin²

2014

Automata, Languages, and Programming

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We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NPhard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of so-called families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NP-hard problems as well as to simplify the proofs of several known results.For the chromatic number problem we present an algorithm with O * ((2 − ε(d)) n ) time and exponential space for graphs of average degree d.

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A Permanent Algorithm with exp[Ω (n ^{1/3}/2 ln n )] Expected Speedup for 0-1 Matrices

Cited by 18 publications

References 8 publications

Computing sparse permanents faster

Computing sparse permanents faster

On Symbolic Approaches for Computing the Matrix Permanent

Families with Infants: A General Approach to Solve Hard Partition Problems

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