A Note on Submodular Function Minimization with Covering Type Linear Constraints

A Note on Submodular Function Minimization with Covering Type Linear Constraints In this paper, we consider the non-negative submodular function minimization problem with covering type linear constraints. Assume that there exist m linear constraints, and we denote by $$\varDelta _i$$ Δ i the number of non-zero coefficients in the ith constraints. Furthermore, we assume that $$\varDelta _1 \ge \varDelta _2 \ge \cdots \ge \varDelta _m$$ Δ 1 ≥ Δ 2 ≥ ⋯ ≥ Δ m . For this problem, Koufogiannakis and Young proposed a polynomial-time $$\varDelta _1$$ Δ 1 -approximation algorithm. In this paper, we propose a new polynomial-time primal-dual approximation algorithm based on the approximation algorithm of Takazawa and Mizuno for the covering integer program with $$\{0,1\}$$ { 0 , 1 } -variables and the approximation algorithm of Iwata and Nagano for the submodular function minimization problem with set covering constraints. The approximation ratio of our algorithm is $$\max \{\varDelta _2, \min \{\varDelta _1, 1 + \varPi \}\}$$ max { Δ 2 , min { Δ 1 , 1 + Π } } , where $$\varPi $$ Π is the maximum size of a connected component of the input submodular function. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algorithmica Springer Journals

A Note on Submodular Function Minimization with Covering Type Linear Constraints

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer Science+Business Media, LLC
Subject
Computer Science; Algorithm Analysis and Problem Complexity; Theory of Computation; Mathematics of Computing; Algorithms; Computer Systems Organization and Communication Networks; Data Structures, Cryptology and Information Theory
ISSN
0178-4617
eISSN
1432-0541
D.O.I.
10.1007/s00453-017-0363-8
Publisher site
See Article on Publisher Site

Abstract

In this paper, we consider the non-negative submodular function minimization problem with covering type linear constraints. Assume that there exist m linear constraints, and we denote by $$\varDelta _i$$ Δ i the number of non-zero coefficients in the ith constraints. Furthermore, we assume that $$\varDelta _1 \ge \varDelta _2 \ge \cdots \ge \varDelta _m$$ Δ 1 ≥ Δ 2 ≥ ⋯ ≥ Δ m . For this problem, Koufogiannakis and Young proposed a polynomial-time $$\varDelta _1$$ Δ 1 -approximation algorithm. In this paper, we propose a new polynomial-time primal-dual approximation algorithm based on the approximation algorithm of Takazawa and Mizuno for the covering integer program with $$\{0,1\}$$ { 0 , 1 } -variables and the approximation algorithm of Iwata and Nagano for the submodular function minimization problem with set covering constraints. The approximation ratio of our algorithm is $$\max \{\varDelta _2, \min \{\varDelta _1, 1 + \varPi \}\}$$ max { Δ 2 , min { Δ 1 , 1 + Π } } , where $$\varPi $$ Π is the maximum size of a connected component of the input submodular function.

Journal

AlgorithmicaSpringer Journals

Published: Aug 14, 2017

References

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