Approximating Submodular Functions Everywhere
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Goemans_Approximating submodular.pdf
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Author(s)
Goemans, Michel X.
Harvey, Nicholas J. A.
Iwata, Satoru
Mirrokni, Vahab
Date Issued
January 2009
Journal
Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2009 (SODA'09)
Publisher
Society for Industrial and Applied Mathematics
Citation
Goemans, Michel X. et al. "Approximating Submodular Functions Everywhere." ACM-SIAM Symposium on Discrete Algorithms, Jan. 4-6, 2009, New York, NY. © 2009 Society for Industrial and Applied Mathematics.
Version
Final published version
Abstract
Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by
a (value) oracle. Many interesting problems involving
submodular functions can be solved using only polynomially many queries to the oracle, e.g., exact minimization or approximate maximization. In this paper, we consider the problem of approximating a non-negative, monotone, submodular function
f on a ground set of size n everywhere, after only poly(n)
oracle queries. Our main result is a deterministic algorithm that makes poly(n) oracle queries and derives a function ^ f such that, for every set S, ^ f(S) approximates f(S) within a factor alpha(n), where alpha(n) = [sqrt]n + 1
for rank functions of matroids and alpha(n) = O( [sqrt]n log n)
for general monotone submodular functions. Our result
is based on approximately finding a maximum volume
inscribed ellipsoid in a symmetrized polymatroid, and
the analysis involves various properties of submodular
functions and polymatroids. Our algorithm is tight up to logarithmic factors.
Indeed, we show that no algorithm can achieve a factor
better than Omega([sqrt]n= log n), even for rank functions of a
matroid.
Description
URL to paper from conference site
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
http://www.siam.org/proceedings/soda/2009/soda09.php
