arXiv subjects

Nima Anari

Publications and source records attributed to Nima Anari.

51 records · Page 3Linked to original sources

arXiv2019

Log-Concave Polynomials II: High-Dimensional Walks and an FPRAS for Counting Bases of a Matroid

We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where $0<q<1$. Consequently, we can sample random spanning forests in a graph and (approximately) compute the reliability polynomial of any matroid. We also prove the thirty year old conjecture of Mihail and Vazirani that the bases exchange graph of any matroid has expansion at least 1. One of our key observations is a close connection between pure simplicial complexes and multiaffine homogeneous polynomials. Specifically, if $X$ is a pure simplicial complex with positive weights on its maximal faces, we can associate with $X$ a multiaffine homogeneous polynomial $p_{X}$ such that the eigenvalues of the localized random walks on $X$ correspond to the eigenvalues of the Hessian of derivatives of $p_{X}$.

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arXiv2018

Log-Concave Polynomials I: Entropy and a Deterministic Approximation Algorithm for Counting Bases of Matroids

We give a deterministic polynomial time $2^{O(r)}$-approximation algorithm for the number of bases of a given matroid of rank $r$ and the number of common bases of any two matroids of rank $r$. To the best of our knowledge, this is the first nontrivial deterministic approximation algorithm that works for arbitrary matroids. Based on a lower bound of Azar, Broder, and Frieze [ABF94] this is almost the best possible result assuming oracle access to independent sets of the matroid. There are two main ingredients in our result: For the first, we build upon recent results of Adiprasito, Huh, and Katz [AHK15] and Huh and Wang [HW17] on combinatorial hodge theory to derive a connection between matroids and log-concave polynomials. We expect that several new applications in approximation algorithms will be derived from this connection in future. Formally, we prove that the multivariate generating polynomial of the bases of any matroid is log-concave as a function over the positive orthant. For the second ingredient, we develop a general framework for approximate counting in discrete problems, based on convex optimization. The connection goes through subadditivity of the entropy. For matroids, we prove that an approximate superadditivity of the entropy holds by relying on the log-concavity of the corresponding polynomials.

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arXiv2018

Log-Concave Polynomials III: Mason's Ultra-Log-Concavity Conjecture for Independent Sets of Matroids

We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.

math.CO↗

arXiv2018

Smoothed Analysis of Discrete Tensor Decomposition and Assemblies of Neurons

We analyze linear independence of rank one tensors produced by tensor powers of randomly perturbed vectors. This enables efficient decomposition of sums of high-order tensors. Our analysis builds upon [BCMV14] but allows for a wider range of perturbation models, including discrete ones. We give an application to recovering assemblies of neurons. Assemblies are large sets of neurons representing specific memories or concepts. The size of the intersection of two assemblies has been shown in experiments to represent the extent to which these memories co-occur or these concepts are related; the phenomenon is called association of assemblies. This suggests that an animal's memory is a complex web of associations, and poses the problem of recovering this representation from cognitive data. Motivated by this problem, we study the following more general question: Can we reconstruct the Venn diagram of a family of sets, given the sizes of their $\ell$-wise intersections? We show that as long as the family of sets is randomly perturbed, it is enough for the number of measurements to be polynomially larger than the number of nonempty regions of the Venn diagram to fully reconstruct the diagram.

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arXiv2018

Planar Graph Perfect Matching is in NC

Is perfect matching in NC? That is, is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in theoretical computer science for over three decades, ever since the discovery of RNC matching algorithms. Within this question, the case of planar graphs has remained an enigma: On the one hand, counting the number of perfect matchings is far harder than finding one (the former is #P-complete and the latter is in P), and on the other, for planar graphs, counting has long been known to be in NC whereas finding one has resisted a solution. In this paper, we give an NC algorithm for finding a perfect matching in a planar graph. Our algorithm uses the above-stated fact about counting matchings in a crucial way. Our main new idea is an NC algorithm for finding a face of the perfect matching polytope at which $Ω(n)$ new conditions, involving constraints of the polytope, are simultaneously satisfied. Several other ideas are also needed, such as finding a point in the interior of the minimum weight face of this polytope and finding a balanced tight odd set in NC.

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arXiv2017

Graph Clustering using Effective Resistance

$ \def\vecc#1{\boldsymbol{#1}} $We design a polynomial time algorithm that for any weighted undirected graph $G = (V, E,\vecc w)$ and sufficiently large $δ> 1$, partitions $V$ into subsets $V_1, \ldots, V_h$ for some $h\geq 1$, such that $\bullet$ at most $δ^{-1}$ fraction of the weights are between clusters, i.e. \[ w(E - \cup_{i = 1}^h E(V_i)) \lesssim \frac{w(E)}δ;\] $\bullet$ the effective resistance diameter of each of the induced subgraphs $G[V_i]$ is at most $δ^3$ times the average weighted degree, i.e. \[ \max_{u, v \in V_i} \mathsf{Reff}_{G[V_i]}(u, v) \lesssim δ^3 \cdot \frac{|V|}{w(E)} \quad \text{ for all } i=1, \ldots, h.\] In particular, it is possible to remove one percent of weight of edges of any given graph such that each of the resulting connected components has effective resistance diameter at most the inverse of the average weighted degree. Our proof is based on a new connection between effective resistance and low conductance sets. We show that if the effective resistance between two vertices $u$ and $v$ is large, then there must be a low conductance cut separating $u$ from $v$. This implies that very mildly expanding graphs have constant effective resistance diameter. We believe that this connection could be of independent interest in algorithm design.

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arXiv2017

Approximating the Largest Root and Applications to Interlacing Families

We study the problem of approximating the largest root of a real-rooted polynomial of degree $n$ using its top $k$ coefficients and give nearly matching upper and lower bounds. We present algorithms with running time polynomial in $k$ that use the top $k$ coefficients to approximate the maximum root within a factor of $n^{1/k}$ and $1+O(\tfrac{\log n}{k})^2$ when $k\leq \log n$ and $k>\log n$ respectively. We also prove corresponding information-theoretic lower bounds of $n^{Ω(1/k)}$ and $1+Ω\left(\frac{\log \frac{2n}{k}}{k}\right)^2$, and show strong lower bounds for noisy version of the problem in which one is given access to approximate coefficients. This problem has applications in the context of the method of interlacing families of polynomials, which was used for proving the existence of Ramanujan graphs of all degrees, the solution of the Kadison-Singer problem, and bounding the integrality gap of the asymmetric traveling salesman problem. All of these involve computing the maximum root of certain real-rooted polynomials for which the top few coefficients are accessible in subexponential time. Our results yield an algorithm with the running time of $2^{\tilde O(\sqrt[3]n)}$ for all of them.

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arXiv2017

Simply Exponential Approximation of the Permanent of Positive Semidefinite Matrices

We design a deterministic polynomial time $c^n$ approximation algorithm for the permanent of positive semidefinite matrices where $c=e^{γ+1}\simeq 4.84$. We write a natural convex relaxation and show that its optimum solution gives a $c^n$ approximation of the permanent. We further show that this factor is asymptotically tight by constructing a family of positive semidefinite matrices.

math.CO↗

arXiv2017

Nash Social Welfare for Indivisible Items under Separable, Piecewise-Linear Concave Utilities

Recently Cole and Gkatzelis gave the first constant factor approximation algorithm for the problem of allocating indivisible items to agents, under additive valuations, so as to maximize the Nash Social Welfare. We give constant factor algorithms for a substantial generalization of their problem -- to the case of separable, piecewise-linear concave utility functions. We give two such algorithms, the first using market equilibria and the second using the theory of stable polynomials. In AGT, there is a paucity of methods for the design of mechanisms for the allocation of indivisible goods and the result of Cole and Gkatzelis seemed to be taking a major step towards filling this gap. Our result can be seen as another step in this direction.

cs.GT↗

arXiv2017

A Generalization of Permanent Inequalities and Applications in Counting and Optimization

A polynomial $p\in\mathbb{R}[z_1,\dots,z_n]$ is real stable if it has no roots in the upper-half complex plane. Gurvits's permanent inequality gives a lower bound on the coefficient of the $z_1z_2\dots z_n$ monomial of a real stable polynomial $p$ with nonnegative coefficients. This fundamental inequality has been used to attack several counting and optimization problems. Here, we study a more general question: Given a stable multilinear polynomial $p$ with nonnegative coefficients and a set of monomials $S$, we show that if the polynomial obtained by summing up all monomials in $S$ is real stable, then we can lowerbound the sum of coefficients of monomials of $p$ that are in $S$. We also prove generalizations of this theorem to (real stable) polynomials that are not multilinear. We use our theorem to give a new proof of Schrijver's inequality on the number of perfect matchings of a regular bipartite graph, generalize a recent result of Nikolov and Singh, and give deterministic polynomial time approximation algorithms for several counting problems.

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arXiv2016

Nash Social Welfare, Matrix Permanent, and Stable Polynomials

We study the problem of allocating $m$ items to $n$ agents subject to maximizing the Nash social welfare (NSW) objective. We write a novel convex programming relaxation for this problem, and we show that a simple randomized rounding algorithm gives a $1/e$ approximation factor of the objective. Our main technical contribution is an extension of Gurvits's lower bound on the coefficient of the square-free monomial of a degree $m$-homogeneous stable polynomial on $m$ variables to all homogeneous polynomials. We use this extension to analyze the expected welfare of the allocation returned by our randomized rounding algorithm.

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arXiv2016

Monte Carlo Markov Chain Algorithms for Sampling Strongly Rayleigh Distributions and Determinantal Point Processes

Strongly Rayleigh distributions are natural generalizations of product and determinantal probability distributions and satisfy strongest form of negative dependence properties. We show that the "natural" Monte Carlo Markov Chain (MCMC) is rapidly mixing in the support of a {\em homogeneous} strongly Rayleigh distribution. As a byproduct, our proof implies Markov chains can be used to efficiently generate approximate samples of a $k$-determinantal point process. This answers an open question raised by Deshpande and Rademacher.

cs.LG↗

arXiv2015

Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP

We show that the integrality gap of the natural LP relaxation of the Asymmetric Traveling Salesman Problem is $\text{polyloglog}(n)$. In other words, there is a polynomial time algorithm that approximates the value of the optimum tour within a factor of $\text{polyloglog}(n)$, where $\text{polyloglog}(n)$ is a bounded degree polynomial of $\log\log(n)$. We prove this by showing that any $k$-edge-connected unweighted graph has a $\text{polyloglog}(n)/k$-thin spanning tree. Our main new ingredient is a procedure, albeit an exponentially sized convex program, that "transforms" graphs that do not admit any spectrally thin trees into those that provably have spectrally thin trees. More precisely, given a $k$-edge-connected graph $G=(V,E)$ where $k\geq 7\log(n)$, we show that there is a matrix $D$ that "preserves" the structure of all cuts of $G$ such that for a set $F\subseteq E$ that induces an $Ω(k)$-edge-connected graph, the effective resistance of every edge in $F$ w.r.t. $D$ is at most $\text{polylog}(k)/k$. Then, we use a recent extension of the seminal work of Marcus, Spielman, and Srivastava [MSS13] by the authors [AO14] to prove the existence of a $\text{polylog}(k)/k$-spectrally thin tree with respect to $D$. Such a tree is $\text{polylog}(k)/k$-combinatorially thin with respect to $G$ as $D$ preserves the structure of cuts of $G$.

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arXiv2015

The Kadison-Singer Problem for Strongly Rayleigh Measures and Applications to Asymmetric TSP

Marcus, Spielman, and Srivastava in their seminal work \cite{MSS13} resolved the Kadison-Singer conjecture by proving that for any set of finitely supported independently distributed random vectors $v_1,\dots, v_n$ which have "small" expected squared norm and are in isotropic position (in expectation), there is a positive probability that the sum $\sum v_i v_i^\intercal$ has small spectral norm. Their proof crucially employs real stability of polynomials which is the natural generalization of real-rootedness to multivariate polynomials. Strongly Rayleigh distributions are families of probability distributions whose generating polynomials are real stable \cite{BBL09}. As independent distributions are just special cases of strongly Rayleigh measures, it is a natural question to see if the main theorem of \cite{MSS13} can be extended to families of vectors assigned to the elements of a strongly Rayleigh distribution. In this paper we answer this question affirmatively; we show that for any homogeneous strongly Rayleigh distribution where the marginal probabilities are upper bounded by $ε_1$ and any isotropic set of vectors assigned to the underlying elements whose norms are at most $\sqrt{ε_2}$, there is a set in the support of the distribution such that the spectral norm of the sum of the natural quadratic forms of the vectors assigned to the elements of the set is at most $O(ε_1+ε_2)$. We employ our theorem to provide a sufficient condition for the existence of spectrally thin trees. This, together with a recent work of the authors \cite{AO14}, provides an improved upper bound on the integrality gap of the natural LP relaxation of the Asymmetric Traveling Salesman Problem.

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arXiv2014

Mechanism Design for Crowdsourcing: An Optimal 1-1/e Competitive Budget-Feasible Mechanism for Large Markets

In this paper we consider a mechanism design problem in the context of large-scale crowdsourcing markets such as Amazon's Mechanical Turk, ClickWorker, CrowdFlower. In these markets, there is a requester who wants to hire workers to accomplish some tasks. Each worker is assumed to give some utility to the requester. Moreover each worker has a minimum cost that he wants to get paid for getting hired. This minimum cost is assumed to be private information of the workers. The question then is - if the requester has a limited budget, how to design a direct revelation mechanism that picks the right set of workers to hire in order to maximize the requester's utility. We note that although the previous work has studied this problem, a crucial difference in which we deviate from earlier work is the notion of large-scale markets that we introduce in our model. Without the large market assumption, it is known that no mechanism can achieve an approximation factor better than 0.414 and 0.5 for deterministic and randomized mechanisms respectively (while the best known deterministic and randomized mechanisms achieve an approximation ratio of 0.292 and 0.33 respectively). In this paper, we design a budget-feasible mechanism for large markets that achieves an approximation factor of 1-1/e (i.e. almost 0.63). Our mechanism can be seen as a generalization of an alternate way to look at the proportional share mechanism which is used in all the previous works so far on this problem. Interestingly, we also show that our mechanism is optimal by showing that no truthful mechanism can achieve a factor better than 1-1/e; thus, fully resolving this setting. Finally we consider the more general case of submodular utility functions and give new and improved mechanisms for the case when the markets are large.

cs.GT↗