A Tight Combinatorial Algorithm for Submodular Maximization Subject to a Matroid Constraint

We present an optimal, combinatorial 1-1/e approximation algorithm for monotone sub modular optimization over a matroid constraint. Compared to the continuous greedy algorithm (Calinescu, Chekuri, Pal and Vondrak, 2008), our algorithm is extremely simple and requires no rounding. It consists of the...

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Bibliographic Details
Published in:2012 IEEE 53rd Annual Symposium on Foundations of Computer Science pp. 659 - 668
Main Authors: Filmus, Y., Ward, J.
Format: Conference Proceeding
Language:English
Published: IEEE 01.10.2012
Subjects:
Algorithm design and analysis
Approximation algorithms
Approximation methods
Greedy algorithms
Linear programming
local search
matroids
Search problems
submodular functions
ISBN:1467343838, 9781467343831
ISSN:0272-5428
Online Access:Get full text
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Summary:We present an optimal, combinatorial 1-1/e approximation algorithm for monotone sub modular optimization over a matroid constraint. Compared to the continuous greedy algorithm (Calinescu, Chekuri, Pal and Vondrak, 2008), our algorithm is extremely simple and requires no rounding. It consists of the greedy algorithm followed by local search. Both phases are run not on the actual objective function, but on a related non-oblivious potential function, which is also monotone sub modular. In our previous work on maximum coverage (Filmus and Ward, 2011), the potential function gives more weight to elements covered multiple times. We generalize this approach from coverage functions to arbitrary monotone sub modular functions. When the objective function is a coverage function, both definitions of the potential function coincide. The parameters used to define the potential function are closely related to Pade approximants of exp(x) evaluated at x = 1. We use this connection to determine the approximation ratio of the algorithm.
ISBN:1467343838
9781467343831
ISSN:0272-5428
DOI:10.1109/FOCS.2012.55