dc.contributor.author | Anari, Nima | |
dc.date.accessioned | 2018-05-17T19:24:03Z | |
dc.date.available | 2018-05-17T19:24:03Z | |
dc.date.issued | 2018-04-30 | |
dc.identifier.uri | http://hdl.handle.net/1853/59687 | |
dc.description | Presented on April 30, 2018 at 12:00 p.m. in the Klaus Advanced Computing Building, Room 1116E. | en_US |
dc.description | Nima Anari is a researcher in the Computer Science Theory Group at Stanford University. He is broadly interested in the design and analysis of algorithms and more generally theoretical computer science. | en_US |
dc.description | Runtime: 58:32 minutes | en_US |
dc.description.abstract | We give a deterministic 2^O(rank) approximation algorithm to count the number of bases of a given matroid and the number of common bases of any two matroids. Based on a lower bound of Azar et al., this is almost the best possible result assuming oracle access to independent sets of matroids.
There are two main ingredients in our result: For the first ingredient, we build upon recent results of Huh et al. and Adiprasito et al. 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 use a general framework for approximate counting in discrete problems, based on convex optimization and sub-additivity of the entropy. For matroids, we prove that an approximate super-additivity of the entropy holds, yielding an approximation algorithm, by relying on log-concavity of the corresponding polynomials. | en_US |
dc.format.extent | 58:32 minutes | |
dc.language.iso | en_US | en_US |
dc.relation.ispartofseries | ARC Colloquium | en_US |
dc.subject | Entropy | en_US |
dc.subject | Matroids | en_US |
dc.title | Entropy, Log-Concavity, and a Deterministic Approximation Algorithm for Counting Bases of Matroids | en_US |
dc.type | Lecture | en_US |
dc.type | Video | en_US |
dc.contributor.corporatename | Georgia Institute of Technology. Algorithms, Randomness and Complexity Center | en_US |
dc.contributor.corporatename | Stanford University. Dept. of Computer Science | en_US |