25 research outputs found
Enumerating contingency tables via random permanents
Given m positive integers R=(r_i), n positive integers C=(c_j) such that sum
r_i = sum c_j =N, and mn non-negative weights W=(w_{ij}), we consider the total
weight T=T(R, C; W) of non-negative integer matrices (contingency tables)
D=(d_{ij}) with the row sums r_i, column sums c_j, and the weight of D equal to
prod w_{ij}^{d_{ij}}. We present a randomized algorithm of a polynomial in N
complexity which computes a number T'=T'(R,C; W) such that T' < T < alpha(R, C)
T' where alpha(R,C) = min{prod r_i! r_i^{-r_i}, prod c_j! c_j^{-c_j}} N^N/N!.
In many cases, ln T' provides an asymptotically accurate estimate of ln T. The
idea of the algorithm is to express T as the expectation of the permanent of an
N x N random matrix with exponentially distributed entries and approximate the
expectation by the integral T' of an efficiently computable log-concave
function on R^{mn}. Applications to counting integer flows in graphs are also
discussed.Comment: 19 pages, bounds are sharpened, references are adde
Notes on the proof of the van der Waerden permanent conjecture
The permanent of an matrix with real entries is defined by the sum where denotes the symmetric group on the -element set . In this creative component we survey some known properties of permanents, calculation of permanents for particular types of matrices and their applications in combinatorics and linear algebra. Then we follow the lines of van Lint\u27s exposition of Egorychev\u27s proof for the van der Waerden\u27s conjecture on the permanents of doubly stochastic matrices. The purpose of this component is to provide elementary proofs of several interesting known facts related to permanents of some special matrices. It is an expository survey paper in nature and reports no new findings
Discrete Mathematics and Symmetry
Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group
Computing solution space properties of combinatorial optimization problems via generic tensor networks
We introduce a unified framework to compute the solution space properties of
a broad class of combinatorial optimization problems. These properties include
finding one of the optimum solutions, counting the number of solutions of a
given size, and enumeration and sampling of solutions of a given size. Using
the independent set problem as an example, we show how all these solution space
properties can be computed in the unified approach of generic tensor networks.
We demonstrate the versatility of this computational tool by applying it to
several examples, including computing the entropy constant for hardcore lattice
gases, studying the overlap gap properties, and analyzing the performance of
quantum and classical algorithms for finding maximum independent sets.Comment: Github repo:
https://github.com/QuEraComputing/GenericTensorNetworks.j