A polynomial-time approximation algorithm for the permanent of a matrix with  non-negative entries

Jerrum, Mark; Sinclair, Alistair; Vigoda, Eric

doi:10.1145/380752.380877

Cited by 165 publications

(283 citation statements)

References 12 publications

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“…Let T ⊆ [n] be defined as in section (2). For each subset S ⊆ T , check if j∈S a ij + b i = 0 for some i = 1, .…”

Section: Avoiding Zero Value Termsmentioning

confidence: 99%

“…To bound the running time of this algorithm, we first note that we must check at most 2 2Cm subsets S of T. Since any term (2) that we add to the running total must have nonzero row sums for each of the first m rows, by Lemma 2.1 there are at most 2 n 1 − 1 2 2C m subsets S = S ∪ R (across all S ⊆ T ) for which we compute (2). Taking m = n 8C , the runtime of this phase of the algorithm is 2 n/4 + poly(n) · 2 n (1 − 1 2 2C ) n/(8C) = 2 n/4 + (2 − ) n for some > 0.…”

Section: Avoiding Zero Value Termsmentioning

confidence: 99%

“…Unfortunately, computing the permanent is believed to be quite hard, as Valiant proved in 1979 that even the problem when restricted to 0/1 matrices is #P complete [4]. On the other hand, a fully-polynomial randomized approximation scheme for computing the permanent of arbitrary matrices with non-negative entries was recently discovered by Jerrum, Sinclair, and Vigoda in [2].…”

Section: Introductionmentioning

confidence: 99%

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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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“…Let T ⊆ [n] be defined as in section (2). For each subset S ⊆ T , check if j∈S a ij + b i = 0 for some i = 1, .…”

Section: Avoiding Zero Value Termsmentioning

confidence: 99%

Section: Avoiding Zero Value Termsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computing sparse permanents faster

Servedio

Wan

2005

Information Processing Letters

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“…Also in computer science, random walks have been applied to various problems such as 2-SAT, approximation of the permanent [10,11], and estimation of the volume of convex bodies [6]. Schöning's elegant algorithm for 3-SAT [15] and its improvement [9] are also based on classical random walks.…”

Section: Introductionmentioning

confidence: 99%

Analysis of absorbing times of quantum walks

2003

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“…As algorithmic tools, they have been applied to a variety of central problems [32], such as estimation of the volume of a convex body [33], approximation of the permanent of a matrix [34], and discovery of satisfying assignments for Boolean formulae [35]. They provide a general paradigm for sampling and exploring an exponentially large set of combinatorial structures (such as matchings in a graph), by using a sequence of simple, local transitions [32].…”

Section: Random Walks In Computer Sciencementioning

confidence: 99%

Quantum random walks: an introductory overview

Kempe¹

2009

Contemporary Physics

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This article aims to provide an introductory survey on quantum random walks. Starting from a physical effect to illustrate the main ideas we will introduce quantum random walks, review some of their properties and outline their striking differences to classical walks. We will touch upon both physical effects and computer science applications, introducing some of the main concepts and language of present day quantum information science in this context. We will mention recent developments in this new area and outline some open questions. God does not play dice.Albert Einstein I. OVERVIEWEver since the discovery of quantum mechanics people have been puzzled by the counter-intuitive character of the laws of nature. Over time we have learned to accept more and more effects that are unimaginable in a classical Newtonian world. Modern technology exploits quantum effects both to our benefit and detriment -among the memorable examples we should cite laser technology and not omit the atomic bomb.In recent years interest in quantum information theory has been generated by the prospect of employing its laws to design devices of surprising power [1]. New ideas include quantum cryptography [2,3] and quantum computation. In 1994 Peter Shor [4] discovered a quantum algorithm to factor numbers efficiently (that is in time that grows only polynomial with the length of the number to be factored), which has unleashed a wave of activity across a broad range of disciplines: physics, computer science, mathematics and engineering.This fruitful axis of research has uncovered many new effects that are strikingly different from their classical counterparts, both from the physical point of view as well as from a computer science and communication theory perspective. Over time these communities have gained a greater understanding of the concepts and notions of the other. The idea that information cannot be separated from the physical devise that is carrying it ('Information is physical') has firmly settled among them and has led to fascinating new insights. Acquaintance with the basic notions of each of these fields seems instrumental in the understanding of modern quantum information processing.In this article we will follow the trajectory of one of the many surprising aspects of quantum information; it is dedicated to quantum random walks. We will give a thorough introduction to the necessary terminology without overburdening the reader with unnecessary mathematics. Starting with a very intuitive and physical example we will step by step introduce the language and notation of present day quantum information science. We will present the necessary background from computer science needed for a physicist to understand and appreciate the developments and results, assuming some rudimentary background of quantum mechanics, but no knowledge of computer science or quantum information theory. An excellent comprehensive introduction to quantum information and computation can be found in [1].In this journey at the interface of several traditional dis...

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A polynomial-time approximation algorithm for the permanent of a matrix with non-negative entries

Cited by 165 publications

References 12 publications

Computing sparse permanents faster

Computing sparse permanents faster

Analysis of absorbing times of quantum walks

Quantum random walks: an introductory overview

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