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

Nima Anari and Kuikui Liu and Shayan Oveis Gharan and Cynthia Vinzant
2019 arXiv   pre-print
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  ...  Consequently, we can sample random spanning forests in a graph and (approximately) compute the reliability polynomial of any matroid.  ...  By the equivalence between approximate counting and approximate sampling for self-reducible problems [JVV86] , this gives an FPRAS for each of the following: 1. counting the bases of a matroid, and 2.  ... 
arXiv:1811.01816v3 fatcat:us52gnv3izeflbipm57fuy5swe

Maximizing Determinants under Matroid Constraints [article]

Vivek Madan, Aleksandar Nikolov, Mohit Singh, Uthaipon Tantipongpipat
2020 arXiv   pre-print
To show the approximation guarantee, we utilize recent works on strongly log-concave polynomials and show new relationships between different convex programs studied for the problem.  ...  The current best results include an e^2k-estimation for any matroid of rank k and a (1+ϵ)^d-approximation for a uniform matroid of rank k> d+d/ϵ, where the rank k> d denotes the desired size of the optimal  ...  Log-concave polynomials ii: high-dimensional walks and an fpras for counting bases of a matroid. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 1-12.  ... 
arXiv:2004.07886v1 fatcat:2z4i3t4wujggvpwd4lst7m6uf4

Approximation Algorithms for NP-Hard Problems

Ravindran Kannan, Marek Karpinski
2004 Oberwolfach Reports  
Thanks to everybody who contributed to the success of this meeting and made it such an enjoyable event!  ...  An important mean for surmounting this intractability barrier is that of approximate computation, where the answer is guaranteed to be within some small fraction of optimality.  ...  Examples of families X for which such an oracle is readily available through existing optimization algorithms include the family of bases of a matroid or bases in the intersection of two matroids on the  ... 
doi:10.4171/owr/2004/28 fatcat:fwbs36pgpjev5gk6cfeb7ylukm

Hypercontractivity on High Dimensional Expanders: Approximate Efron-Stein Decompositions for ε-Product Spaces [article]

Tom Gur, Noam Lifshitz, Siqi Liu
2021 arXiv   pre-print
Our techniques rely on a new approximate Efron-Stein decomposition for high dimensional link expanders.  ...  As applications, we obtain Fourier concentration, small-set expansion, and Kruskal-Katona theorems for high dimensional expanders.  ...  [4] Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials ii: high-dimensional walks and an fpras for counting bases of a matroid.  ... 
arXiv:2111.09375v4 fatcat:pd4s5ihsqjcdvox5sikcrqmaoy

Markov chains and polynomial time algorithms

R. Kannan
Proceedings 35th Annual Symposium on Foundations of Computer Science  
They fall into two classes : combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing the volumes of convex sets.  ...  This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting problems.  ...  Acknowledgement : Many thanks to Alan Frieze and John Mount for reading the manuscript and suggesting many improvements.  ... 
doi:10.1109/sfcs.1994.365726 dblp:conf/focs/Kannan94 fatcat:qoereei67je7fe5jec5gpierdu

Scientific Visualization (Dagstuhl Seminar 11231) Outdoor and Large-Scale Real-World Scene Analysis. 15th Workshop Theoretic Foundations of Computer Vision

Min Chen, Hans Hagen, Charles Hansen, Arie Kaufman, Martin Dyer, Uriel Feige, Alan Frieze, Marek Karpinski, Frank Dellaert, Jan-Michael Frahm, Marc Pollefeys, Bodo Rosenhahn
2011 unpublished
Thanks go to Mathias Hauptmann for his help in collecting abstracts of the talks and other related materials for these Proceedings.  ...  We thank Annette Beyer and Angelika Mueller-von Brochowski for their continuous support and help in organizing this workshop.  ...  Jerrum and Sinclair have shown that there is a fully polynomial randomised approximation scheme (FPRAS) for the class of graphic matroids.  ... 

0/1-Polytopes: Typical and Extremal Properties [article]

Rafael Gillmann, Technische Universität Berlin, Technische Universität Berlin, Volker Kaibel
After a short introduction to the basics of polytope theory and an survey on the state-of-the-art on 0/1-polytopes, a short introduction to probability theory is given.  ...  face; (b) there are 0/1-polytopes with exponentially small vertex expansion; (c) for every $k$ there is a constant $c_k$ such that a random $d$-dimensional 0/1-polytope with at most $2^{c_kd}$ vertices  ...  For example, it would yield an ecient algorithm to approximate the number of bases of a matroid.  ... 
doi:10.14279/depositonce-1526 fatcat:akxt3czuk5duloxopwde36kuay