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

Jerrum, Mark; Sinclair, Alistair; Vigoda, Eric

doi:10.1145/1008731.1008738

Cited by 578 publications

(476 citation statements)

References 17 publications

Supporting

Mentioning

472

Contrasting

Order By: Relevance

“…One considers a graph G on vertex set V so that the limit distribution of the random walk on G is F. A clever choice of G can guarantee that (i) it is feasible to efficiently simulate this random walk and (ii) the distribution induced on V by the walk converges rapidly to F. Among the fields where this methodology plays an important role are Statistical Physics, Computational Group Theory and Combinatorial Optimization. We should mention approximation algorithms for the permanence of nonnegative matrices [JSV04] and for the volume of convex bodies in high dimension [Sim03] as prime examples of the latter. Excellent surveys on the subject are [JS96,Jer03].…”

Section: Random Walks On Expander Graphsmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,470

1,355

View full text Add to dashboard Cite

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields.But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are "just" one-dimensional complexes, they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected. Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early '70s. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness. In mathematics, we will encounter e.g. their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture. Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications. The list of such interesting and fruitful connections goes on and on with so many applications we will not even be able to mention. This universality of expanders is becoming more evident as more connections are discovered. It transpires that expansion is a fundamental mathematical concept, well deserving to be thoroughly investigated on its own.In hindsight, one reason that expanders are so ubiquitous is that their very definition can be given in at least three languages: combinatorial/geometric, probabilistic and algebraic. Combinatorially, expanders are graphs which are highly connected; to disconnect a large part of the graph, one has to sever many edges. Equivalently, using the geometric notion of isoperimetry, every s...

show abstract

Section: Random Walks On Expander Graphsmentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,470

1,355

View full text Add to dashboard Cite

show abstract

“…Important examples are randomized polynomial-time approximation schemes for evaluating the permanent of a nonnegative matrix [1], the volume of a convex polytope [2], and the partition functions of the monomer-dimer and ferromagnetic Ising systems [3,4]. The centerpiece of all these algorithms is the Markov Chain Monte Carlo (MCMC) method, making it possible to approximately sample from a particular probability distribution π over a large set Ω.…”

Section: Introductionmentioning

confidence: 99%

Speedup via quantum sampling

2008

View full text Add to dashboard Cite

The Markov Chain Monte Carlo method is at the heart of efficient approximation schemes for a wide range of problems in combinatorial enumeration and statistical physics. It is therefore very natural and important to determine whether quantum computers can speed-up classical mixing processes based on Markov chains. To this end, we present a new quantum algorithm, making it possible to prepare a quantum sample, i.e., a coherent version of the stationary distribution of a reversible Markov chain. Our algorithm has a significantly better running time than that of a previous algorithm based on adiabatic state generation. We also show that our methods provide a speed-up over a recently proposed method for obtaining ground states of (classical) Hamiltonians.

show abstract

“…Unfortunately, nonnegative permanent computation is #P -hard, as shown by [14]. On the other hand, a groundbreaking result of [11] presents a polynomial time approximation scheme for permanent, which runs in time O(n 10 ) for fixed accuracy. To make things worse, the dependence in the accuracy is inverse polynomial, implying that, even if we could perform arbitrarily accurate floating point operations, the total running time would be super linear in T , because a regret dependence of √ T over T steps requires accuracy inverse polynomial in T .…”

Section: Results Techniques and Contributionmentioning

confidence: 99%

Bandit online optimization over the permutahedron

Ailon

Hatano

Takimoto

2016

Theoretical Computer Science

View full text Add to dashboard Cite

The permutahedron is the convex polytope with vertex set consisting of the vectors (π(1), . . . , π(n)) for all permutations (bijections) π over {1, . . . , n}. We study a bandit game in which, at each step t, an adversary chooses a hidden weight weight vector st, a player chooses a vertex πt of the permutahedron and suffers an observed instantaneous loss ofWe study the problem in two regimes. In the first regime, st is a point in the polytope dual to the permutahedron. Algorithm CombBand of Cesa-Bianchi et al (2009) guarantees a regret of O(n √ T log n) after T steps. Unfortunately, CombBand requires at each step an n-by-n matrix permanent computation, a #P -hard problem. Approximating the permanent is possible in the impractical running time of O(n 10 ), with an additional heavy inverse-polynomial dependence on the sought accuracy. We provide an algorithm of slightly worse regret O(n 3/2 √ T ) but with more realistic time complexity O(n 3 ) per step. The technical contribution is a bound on the variance of the Plackett-Luce noisy sorting process's 'pseudo loss', obtained by establishing positive semi-definiteness of a family of 3-by-3 matrices of rational functions in exponents of 3 parameters.In the second regime, st is in the hypercube. For this case we present and analyze an algorithm based on Bubeck et al.'s (2012) OSMD approach with a novel projection and decomposition technique for the permutahedron. The algorithm is efficient and achieves a regret of O(n √ T ), but for a more restricted space of possible loss vectors.

show abstract

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

Cited by 578 publications

References 17 publications

Expander graphs and their applications

Expander graphs and their applications

Speedup via quantum sampling

Bandit online optimization over the permutahedron

Contact Info

Product

Resources

About