
Approximately counting bases of bicircular matroids
We give a fully polynomialtime randomised approximation scheme (FPRAS) ...
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Maximizing Determinants under Matroid Constraints
Given vectors v_1,…,v_n∈ℝ^d and a matroid M=([n],I), we study the proble...
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LogConcave Polynomials III: Mason's UltraLogConcavity Conjecture for Independent Sets of Matroids
We give a selfcontained proof of the strongest version of Mason's conje...
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Zeros of Holant problems: locations and algorithms
We present fully polynomialtime (deterministic or randomised) approxima...
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Isotropy and LogConcave Polynomials: Accelerated Sampling and HighPrecision Counting of Matroid Bases
We define a notion of isotropy for discrete set distributions. If μ is a...
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LogConcave Polynomials II: HighDimensional Walks and an FPRAS for Counting Bases of a Matroid
We use recent developments in the area of high dimensional expanders and...
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LogConcave Polynomials IV: Exchange Properties, Tight Mixing Times, and Faster Sampling of Spanning Trees
We prove tight mixing time bounds for natural random walks on bases of m...
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Logconcave polynomials, 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 logconcave 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 logconcave 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 logconcavity of the corresponding polynomials.
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