A polynomial time algorithm to approximate the mixed volume within a simply exponential factor

Let ${\bf K} = (K_1, ..., K_n)$ be an $n$-tuple of convex compact subsets in the Euclidean space $\R^n$, and let $V(\cdot)$ be the Euclidean volume in $\R^n$. The Minkowski polynomial $V_{{\bf K}}$ is defined as $V_{{\bf K}}(\lambda_1, ... ,\lambda_n) = V(\lambda_1 K_1 +, ..., + \lambda_n K_n)$ and the mixed volume $V(K_1, ..., K_n)$ as $$V(K_1, ..., K_n) = \frac{\partial^n}{\partial \lambda_1...\partial \lambda_n} V_{{\bf K}}(\lambda_1 K_1 +, ..., + \lambda_n K_n).$$ Our main result is a poly-time algorithm which approximates $V(K_1, ..., K_n)$ with multiplicative error $e^n$ and with better rates if the affine dimensions of most of the sets $K_i$ are small. Our approach is based on a particular approximation of $\log(V(K_1, ..., K_n))$ by a solution of some convex minimization problem. We prove the mixed volume analogues of the Van der Waerden and Schrijver-Valiant conjectures on the permanent. These results, interesting on their own, allow us to justify the abovementioned approximation by a convex minimization, which is solved using the ellipsoid method and a randomized poly-time time algorithm for the approximation of the volume of a convex set.Comment: a journal version, accepted to Discrete and Computational Geometr

Hyperbolic Polynomials Approach to Van der Waerden/Schrijver-Valiant like Conjectures : Sharper Bounds, Simpler Proofs and Algorithmic Applications

Let $p(x_1,...,x_n) = p(X), X \in R^{n}$ be a homogeneous polynomial of degree $n$ in $n$ real variables, $e = (1,1,..,1) \in R^n$ be a vector of all ones . Such polynomial $p$ is called $e$-hyperbolic if for all real vectors $X \in R^{n}$ the univariate polynomial equation $P(te - X) = 0$ has all real roots $\lambda_{1}(X) \geq ... \geq \lambda_{n}(X)$ . The number of nonzero roots $|\{i :\lambda_{i}(X) \neq 0 \}|$ is called $Rank_{p}(X)$ . A $e$-hyperbolic polynomial $p$ is called $POS$-hyperbolic if roots of vectors $X \in R^{n}_{+}$ with nonnegative coordinates are also nonnegative (the orthant $R^{n}_{+}$ belongs to the hyperbolic cone) and $p(e) > 0$ . Below $\{e_1,...,e_n\}$ stands for the canonical orthogonal basis in $R^{n}$. The main results states that if $p(x_1,x_2,...,x_n)$ is a $POS$-hyperbolic (homogeneous) polynomial of degree $n$, $Rank_{p} (e_{i}) = R_i$ and $p(x_1,x_2,...,x_n) \geq \prod_{1 \leq i \leq n} x_i ; x_i > 0, 1 \leq i \leq n ,$ then the following inequality holds $$\frac{\partial^n}{\partial x_1...\partial x_n} p(0,...,0) \geq \prod_{1 \leq i \leq n} (\frac{G_{i} -1}{G_{i}})^{G_{i} -1} (G_i = \min(R_{i}, n+1-i)) .$$ This theorem is a vast (and unifying) generalization of as the van der Waerden conjecture on the permanents of doubly stochastic matrices as well Schrijver-Valiant conjecture on the number of perfect matchings in $k$-regular bipartite graphs . The paper is (almost) self-contained, most of the proofs can be found in the {\bf Appendices}.Comment: 21 pages, some typos are corrected . Section 2.2 is added . This section sketches the extension of the main result to the volume polynomials . This extension leads to a randomized poly-time algorithm to approximate the mixed volume of n compact convex sets in R^n within multiplicative factor e^

Quantum Matching Theory (with new complexity theoretic, combinatorial and topological insights on the nature of the Quantum Entanglement)

Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. Based on this point of view, we introduce a definition of perfect Quantum (operator) matching . We show that the new notion inherits many "classical" properties, but not all of them . This new notion goes somewhere beyound matroids . For separable bipartite quantum states this new notion coinsides with the full rank property of the intersection of two corresponding geometric matroids . In the classical situation, permanents are naturally associated with perfects matchings. We introduce an analog of permanents for positive operators, called Quantum Permanent and show how this generalization of the permanent is related to the Quantum Entanglement. Besides many other things, Quantum Permanents provide new rational inequalities necessary for the separability of bipartite quantum states . Using Quantum Permanents, we give deterministic poly-time algorithm to solve Hidden Matroids Intersection Problem and indicate some "classical" complexity difficulties associated with the Quantum Entanglement. Finally, we prove that the weak membership problem for the convex set of separable bipartite density matrices is NP-HARD.Comment: 10 pages, slightly edited version, proof of hardness of weak membership problem for separability added (last section

Van der Waerden Conjecture for Mixed Discriminants

We prove that the mixed discriminant of doubly stochastic $n$-tuples of semidefinite hermitian $n \times n$ matrices is bounded below by $\frac{n!}{n^{n}}$ and that this bound is uniquely attained at the $n$-tuple $(\frac{1}{n} I,...,\frac{1}{n} I)$. This result settles a conjecture posed by R. Bapat in 1989. We consider various generalizations and applications of this result.Comment: 18 page

Classical deterministic complexity of Edmonds' problem and Quantum Entanglement

This paper continues research initiated in quant-ph/0201022 . The main subject here is the so-called Edmonds' problem of deciding if a given linear subspace of square matrices contains a nonsingular matrix . We present a deterministic polynomial time algorithm to solve this problem for linear subspaces satisfying a special matroids motivated property, called in the paper the Edmonds-Rado property . This property is shown to be very closely related to the separability of bipartite mixed states . One of the main tools used in the paper is the Quantum Permanent introduced in quant-ph/0201022 .Comment: 31 pages, long version of STOC-2003 pape

A proof of hyperbolic van der Waerden conjecture : the right generalization is the ultimate simplification

Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$ complex variables . Assume that this polynomial satisfies the property : \\ $|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain $\{(z_1,...,z_n) : Re(z_i) \geq 0, 1 \leq i \leq n \}$ . \\ We prove that $|\frac{\partial^n}{\partial z_1...\partial z_n} p | \geq \frac{n!}{n^n}$ . Our proof is relatively short and self-contained (i.e. we only use basic properties of hyperbolic polynomials). As the van der Waerden conjecture for permanents, proved by D.I. Falikman and G.P. Egorychev, as well Bapat's conjecture for mixed discriminants, proved by the author, are particular cases of this result. We also prove so called "small rank" lower bound (in the permanents context it corresponds to sparse doubly-stochastic matrices, i.e. with small number of non-zero entries in each column). The later lower bound generalizes (with simpler proofs) recent lower bounds by A.Schrijver for the number of perfect matchings of $k$-regular bipartite graphs. We present some important algorithmic applications of the result, including a polynomial time deterministic algorithm approximating the permanent of $n \times n$ nonnegative entry-wise matrices within multiplicative factor $\frac{e^n}{n^m}$ for any fixed positive $m$ .Comment: 15 pages, preliminary (still) version . A subsection on generalizations (with simpler proofs) of recent lower bounds by A.Schrijver for the number of perfect matchings of $k$-regular bipartite graph

Combinatorial and algorithmic aspects of hyperbolic polynomials

Let $p(x_1,...,x_n) =\sum_{(r_1,...,r_n) \in I_{n,n}} a_{(r_1,...,r_n)} \prod_{1 \leq i \leq n} x_{i}^{r_{i}}$ be homogeneous polynomial of degree $n$ in $n$ real variables with integer nonnegative coefficients. The support of such polynomial $p(x_1,...,x_n)$ is defined as $supp(p) = \{(r_1,...,r_n) \in I_{n,n} : a_{(r_1,...,r_n)} \neq 0 \}$ . The convex hull $CO(supp(p))$ of $supp(p)$ is called the Newton polytope of $p$ . We study the following decision problems, which are far-reaching generalizations of the classical perfect matching problem : {itemize} {\bf Problem 1 .} Consider a homogeneous polynomial $p(x_1,...,x_n)$ of degree $n$ in $n$ real variables with nonnegative integer coefficients given as a black box (oracle) . {\it Is it true that $(1,1,..,1) \in supp(p)$ ?} {\bf Problem 2 .} Consider a homogeneous polynomial $p(x_1,...,x_n)$ of degree $n$ in $n$ real variables with nonnegative integer coefficients given as a black box (oracle) . {\it Is it true that $(1,1,..,1) \in CO(supp(p))$ ?} {itemize} We prove that for hyperbolic polynomials these two problems are equivalent and can be solved by deterministic polynomial-time oracle algorithms . This result is based on a "hyperbolic" generalization of Rado theorem .Comment: 28 pages, extended and edited version . A proof of Conjecture 2.11 (hyperbolic van der Waerden conjecture) will be posted shortl

B\'ezout type inequality in convex geometry

We give a B\'ezout type inequality for mixed volumes, which holds true for any convex bodies. The key ingredient is the reverse Khovanskii-Teissier inequality for convex bodies, which was obtained in our previous work and inspired by its correspondence in complex geometry.Comment: 8 page

Stability and complexity of mixed discriminants

We show that the mixed discriminant of $n$ positive semidefinite $n \times n$ real symmetric matrices can be approximated within a relative error $\epsilon >0$ in quasi-polynomial $n^{O(\ln n -\ln \epsilon)}$ time, provided the distance of each matrix to the identity matrix in the operator norm does not exceed some absolute constant $\gamma_0 >0$. We deduce a similar result for the mixed discriminant of doubly stochastic $n$-tuples of matrices from the Marcus - Spielman - Srivastava bound on the roots of the mixed characteristic polynomial. Finally, we construct a quasi-polynomial algorithm for approximating the sum of $m$-th powers of principal minors of a matrix, provided the operator norm of the matrix is strictly less than 1. As is shown by Gurvits, for $m=2$ the problem is $\#P$-hard and covers the problem of computing the mixed discriminant of positive semidefinite matrices of rank 2.Comment: Included an algorithm for computing the sum of powers of principal minors of a matri

Combinatorics hidden in hyperbolic polynomials and related topics

The main topic of this paper is various "hyperbolic" generalizations of the Edmonds-Rado theorem on the rank of intersection of two matroids. We prove several results in this direction and pose a few questions. We also give generalizations of the Obreschkoff theorem and recent results of J. Borcea and B. Shapiro.Comment: 20 page