arXiv subjects

Alexander Barvinok

Publications and source records attributed to Alexander Barvinok.

At least 37 records · Page 2Linked to original sources

arXiv2016

Computing the partition function of a polynomial on the Boolean cube

For a polynomial f: {-1, 1}^n --> C, we define the partition function as the average of e^{lambda f(x)} over all points x in {-1, 1}^n, where lambda in C is a parameter. We present a quasi-polynomial algorithm, which, given such f, lambda and epsilon >0 approximates the partition function within a relative error of epsilon in N^{O(ln n -ln epsilon)} time provided |lambda| < 1/(2 L sqrt{deg f}), where L=L(f) is a parameter bounding the Lipschitz constant of f from above and N is the number of monomials in f. As a corollary, we obtain a quasi-polynomial algorithm, which, given such an f with coefficients +1 and -1 and such that every variable enters not more than 4 monomials, approximates the maximum of f on {-1, 1}^n within a factor of O(sqrt{deg f}/delta), provided the maximum is N delta for some 0< delta <1. If every variable enters not more than k monomials for some fixed k > 4, we are able to establish a similar result when delta > (k-1)/k.

cs.DS↗

arXiv2015

Computing the partition function for graph homomorphisms with multiplicities

We consider a refinement of the partition function of graph homomorphisms and present a quasi-polynomial algorithm to compute it in a certain domain. As a corollary, we obtain quasi-polynomial algorithms for computing partition functions for independent sets, perfect matchings, Hamiltonian cycles and dense subgraphs in graphs as well as for graph colorings. This allows us to tell apart in quasi-polynomial time graphs that are sufficiently far from having a structure of a given type (i.e., independent set of a given size, Hamiltonian cycle, etc.) from graphs that have sufficiently many structures of that type, even when the probability to hit such a structure at random is exponentially small.

math.CO↗

arXiv2015

Concentration of the mixed discriminant of well-conditioned matrices

We call an n-tuple Q_1, ..., Q_n of positive definite nxn matrices alpha-conditioned for some alpha > 1 if the ratio of the largest among the eigenvalues of Q_1, ..., Q_n to the smallest among the eigenvalues of Q_1, ..., Q_n does not exceed alpha. An n-tuple is called doubly stochastic if the sum of Q_i is the identity matrix and the trace of each Q_i is 1. We prove that for any fixed alpha > 1 the mixed discriminant of an alpha-conditioned doubly stochastic n-tuple is n^{O(1)} e^{-n}. As a corollary, for any alpha > 1 fixed in advance, we obtain a polynomial time algorithm approximating the mixed discriminant of an alpha-conditioned n-tuple within a polynomial in n factor.

math.CO↗

arXiv2015

Computing the partition function for graph homomorphisms

We introduce the partition function of edge-colored graph homomorphisms, of which the usual partition function of graph homomorphisms is a specialization, and present an efficient algorithm to approximate it in a certain domain. Corollaries include efficient algorithms for computing weighted sums approximating the number of k-colorings and the number of independent sets in a graph, as well as an efficient procedure to distinguish pairs of edge-colored graphs with many color-preserving homomorphisms G --> H from pairs of graphs that need to be substantially modified to acquire a color-preserving homomorphism G --> H.

math.CO↗

arXiv2014

Computing the partition function for cliques in a graph

We present a deterministic algorithm which, given a graph G with n vertices and an integer 1 0 is an absolute constant: we can choose gamma=0.06, and if n > 4m and m > 10, we can choose gamma=0.18. This allows us to tell apart the graphs that do not have m-subsets of high density from the graphs that have sufficiently many m-subsets of high density, even when the probability to hit such a subset at random is exponentially small in m.

math.CO↗

arXiv2014

Computing the permanent of (some) complex matrices

We present a deterministic algorithm, which, for any given 0< epsilon < 1 and an nxn real or complex matrix A=(a_{ij}) such that | a_{ij}-1| < 0.19 for all i, j computes the permanent of A within relative error epsilon in n^{O(ln n -ln epsilon)} time. The method can be extended to computing hafnians and multidimensional permanents.

math.CO↗

arXiv2013

Partition functions for dense instances of combinatorial enumeration problems

Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds: cycle covers, closed walks, spanning trees. We show that if the weights of the edges of the graph are within a constant factor, fixed in advance, of each other then the bulk of the partition function is concentrated on the subsets of a particularly simple structure: cycle covers with few cycles, walks that visit every vertex only few times, and spanning trees with small degree of every vertex. This allows us to construct a polynomial time algorithm to separate graphs with many Hamiltonian cycles from graphs that are sufficiently far from Hamiltonian.

math.CO↗

arXiv2013

Convexity of the image of a quadratic map via the relative entropy distance

Let psi: R^n --> R^k be a map defined by k positive definite quadratic forms on R^n. We prove that the relative entropy (Kullback-Leibler) distance from the convex hull of the image of psi to the image of psi is bounded above by an absolute constant. More precisely, we prove that for every point a=(a_1, ..., a_k) in the convex hull of the image of psi such that a_1 + ... +a_k =1 there is a point b=(b_1, ..., b_k) in the image of psi such that b_1 + ... + b_k =1 and such that a_1 ln(a_1/b_1) + ... + a_k ln(a_k/b_k) < 4.8. Similarly, we prove that for any integer m one can choose a convex combination b of at most m points from the image of psi such that a_1 ln(a_1/b_1) + ... + a_k ln(a_k/b_k) < 15/sqrt{m}.

math.MG↗

arXiv2012

Thrifty approximations of convex bodies by polytopes

Given a convex body C in R^d containing the origin in its interior and a real number tau > 1 we seek to construct a polytope P in C with as few vertices as possible such that C in tau P. Our construction is nearly optimal for a wide range of d and tau. In particular, we prove that if C=-C then for any 1>epsilon>0 and tau=1+epsilon one can choose P having roughly epsilon^{-d/2} vertices and for tau=sqrt{epsilon d} one can choose P having roughly d^{1/epsilon} vertices. Similarly, we prove that if C in R^d is a convex body such that -C in mu C for some mu > 1 then one can choose P having roughly ((mu+1)/(tau-1))^{d/2} vertices provided (tau-1)/(mu+1) << 1.

math.MG↗

arXiv2012

A bound for the number of vertices of a polytope with applications

We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has 2^{Omega(n)} vertices and that the number of r-factors in a k-regular graph is exponentially large in the number of vertices of the graph provided k >2r and every cut in the graph with at least two vertices on each side has more than k/r edges.

math.CO↗

arXiv2012

Explicit constructions of centrally symmetric k-neighborly polytopes and large strictly antipodal sets

We present explicit constructions of centrally symmetric 2-neighborly d-dimensional polytopes with about 3^{d/2} = (1.73)^d vertices and of centrally symmetric k-neighborly d-polytopes with about 2^{c_k d} vertices where c_k=3/20 k^2 2^k. Using this result, we construct for a fixed k > 1 and arbitrarily large d and N, a centrally symmetric d-polytope with N vertices that has at least (1-k^2 (gamma_k)^d) binom(N, k) faces of dimension k-1, where gamma_2=1/\sqrt{3} = 0.58 and gamma_k = 2^{-3/{20k^2 2^k}} for k > 2. Another application is a construction of a set of 3^{d/2 -1}-1 points in R^d every two of which are strictly antipodal as well as a construction of an n-point set (for an arbitrarily large n) in R^d with many pairs of strictly antipodal points. The two latter results significantly improve the previous bounds by Talata, and Makai and Martini, respectively.

math.MG↗

arXiv2012

Approximations of convex bodies by polytopes and by projections of spectrahedra

We prove that for any compact set B in R^d and for any epsilon >0 there is a finite subset X of B of |X|=d^{O(1/epsilon^2)} points such that the maximum absolute value of any linear function ell: R^d --> R on X approximates the maximum absolute value of ell on B within a factor of epsilon sqrt{d}. We also discuss approximations of convex bodies by projections of spectrahedra, that is, by projections of sections of the cone of positive semidefinite matrices by affine subspaces.

math.MG↗

arXiv2011

The number of graphs and a random graph with a given degree sequence

We consider the set of all graphs on n labeled vertices with prescribed degrees D=(d_1, ..., d_n). For a wide class of tame degree sequences D we prove a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0-1 matrices with prescribed row and column sums.

math.CO↗

arXiv2011

Centrally symmetric polytopes with many faces

We present explicit constructions of centrally symmetric polytopes with many faces: first, we construct a d-dimensional centrally symmetric polytope P with about (1.316)^d vertices such that every pair of non-antipodal vertices of P spans an edge of P, second, for an integer k>1, we construct a d-dimensional centrally symmetric polytope P of an arbitrarily high dimension d and with an arbitrarily large number N of vertices such that for some 0 < delta_k < 1 at least (1-delta_k^d) {N choose k} k-subsets of the set of vertices span faces of P, and third, for an integer k>1 and a>0, we construct a centrally symmetric polytope Q with an arbitrary large number N of vertices and of dimension d=k^{1+o(1)} such that least (1 - k^{-a}){N choose k} k-subsets of the set of vertices span faces of Q.

math.MG↗

arXiv2011

Neighborliness of the symmetric moment curve

We consider the convex hull B_k of the symmetric moment curve U(t)=(cos t, sin t, cos 3t, sin 3t, ..., cos (2k-1)t, sin (2k-1)t) in R^{2k}, where t ranges over the unit circle S= R/2pi Z. The curve U(t) is locally neighborly: as long as t_1, ..., t_k lie in an open arc of S of a certain length phi_k>0, the convex hull of the points U(t_1), ..., U(t_k) is a face of B_k. We characterize the maximum possible length phi_k, proving, in particular, that phi_k > pi/2 for all k and that the limit of phi_k is pi/2 as k grows. This allows us to construct centrally symmetric polytopes with a record number of faces.

math.MG↗

arXiv2011

Computing the partition function for perfect matchings in a hypergraph

Given non-negative weights w_S on the k-subsets S of a km-element set V, we consider the sum of the products w_{S_1} ... w_{S_m} for all partitions V = S_1 cup ... cup S_m into pairwise disjoint k-subsets S_i. When the weights w_S are positive and within a constant factor, fixed in advance, of each other, we present a simple polynomial time algorithm to approximate the sum within a polynomial in m factor. In the process, we obtain higher-dimensional versions of the van der Waerden and Bregman-Minc bounds for permanents. We also discuss applications to counting of perfect and nearly perfect matchings in hypergraphs.

math.CO↗

arXiv2010

Matrices with prescribed row and column sums

This is a survey of the recent progress and open questions on the structure of the sets of 0-1 and non-negative integer matrices with prescribed row and column sums. We discuss cardinality estimates, the structure of a random matrix from the set, discrete versions of the Brunn-Minkowski inequality and the statistical dependence between row and column sums.

math.CO↗

arXiv2010

Maximum entropy Edgeworth estimates of the number of integer points in polytopes

Abstract: The number of points $x=(x_1 ,x_2 ,...x_n)$ that lie in an integer cube $C$ in $R^n$ and satisfy the constraints $\sum_j h_{ij}(x_j )=s_i ,1\le i\le d$ is approximated by an Edgeworth-corrected Gaussian formula based on the maximum entropy density $p$ on $x \in C$, that satisfies $E\sum_j h_{ij}(x_j )=s_i ,1\le i\le d$. Under $p$, the variables $X_1 ,X_2 ,...X_n $ are independent with densities of exponential form. Letting $S_i$ denote the random variable $\sum_j h_{ij}(X_j )$, conditional on $S=s, X$ is uniformly distributed over the integers in $C$ that satisfy $S=s$. The number of points in $C$ satisfying $S=s$ is $p \{S=s\}\exp (I(p))$ where $I(p)$ is the entropy of the density $p$. We estimate $p \{S=s\}$ by $p_Z(s)$, the density at $s$ of the multivariate Gaussian $Z$ with the same first two moments as $S$; and when $d$ is large we use in addition an Edgeworth factor that requires the first four moments of $S$ under $p$. The asymptotic validity of the Edgeworth-corrected estimate is proved and demonstrated for counting contingency tables with given row and column sums as the number of rows and columns approaches infinity, and demonstrated for counting the number of graphs with a given degree sequence, as the number of vertices approaches infinity.

stat.ME↗