A Faster Scaling Algorithm for Minimizing Submodular Functions

Iwata, Satoru

doi:10.1007/3-540-47867-1_1

Cited by 44 publications

(67 citation statements)

References 11 publications

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“…Combinatorial strongly polynomial algorithms for minimizing submodular functions were developed later by Iwata, Fleischer, and Fujishige [15] and by Schrijver [26]. These combinatorial algorithms have been improved in time complexity [14,16,25].…”

Section: Introductionmentioning

confidence: 99%

Submodular Function Minimization under Covering Constraints

Iwata

Nagano

2009

2009 50th Annual IEEE Symposium on Foundations of Computer Science

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107

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This paper addresses the problems of minimizing nonnegative submodular functions under covering constraints, which generalize the vertex cover, edge cover, and set cover problems. We give approximation algorithms for these problems exploiting the discrete convexity of submodular functions. We first present a rounding 2-approximation algorithm for the submodular vertex cover problem based on the half-integrality of the continuous relaxation problem, and show that the rounding algorithm can be performed by one application of submodular function minimization on a ring family. We also show that a rounding algorithm and a primal-dual algorithm for the submodular cost set cover problem are both constant factor approximation algorithms if the maximum frequency is fixed. In addition, we give an essentially tight lower bound on the approximability of the submodular edge cover problem.

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Section: Introductionmentioning

confidence: 99%

Submodular Function Minimization under Covering Constraints

Iwata

Nagano

2009

2009 50th Annual IEEE Symposium on Foundations of Computer Science

Self Cite

107

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“…Submodular function maximization is easily shown to be NPhard [34] since it generalizes many standard NP-hard problems such as the maximum cut problem [12,9]. In contrast, the problem of minimizing a submodular function can be solved efficiently with only polynomially many evaluations of the function [19] either by using the ellipsoid algorithm [13,14], or by using one of several combinatorial algorithms that have been obtained in the last decade [33,20,17,18,30,22].…”

Section: Introductionmentioning

confidence: 99%

Maximizing Bisubmodular and k-Submodular Functions

Ward¹,

Živný²

2013

Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms

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Submodular functions play a key role in combinatorial optimization and in the study of valued constraint satisfaction problems. Recently, there has been interest in the class of bisubmodular functions, which assign values to disjoint pairs of sets. Like submodular functions, bisubmodular functions can be minimized exactly in polynomial time and exhibit the property of diminishing returns common to many problems in operations research. Recently, the class of k-submodular functions has been proposed. These functions generalize the notion of submodularity to k-tuples of sets, with submodular and bisubmodular functions corresponding to k = 1 and 2, respectively.In this paper, we consider the problem of maximizing bisubmodular and, more generally, k-submodular functions in the value oracle model. We provide the first approximation guarantees for maximizing a general bisubmodular or k-submodular function. We give an analysis of the naive random algorithm as well as a randomized greedy algorithm inspired by the recent randomized greedy algorithm of Buchbinder et al. [FOCS'12] for unconstrained submodular maximization. We show that this algorithm approximates any k-submodular function to a factor of 1/(1 + p k/2). In the case of bisubmodular functions, our randomized greedy algorithm gives an approximation guarantee of 1/2. We show that, as in the case of submodular functions, this result is the best possible in both the value query model, and under the assumption that N P = RP . Our analysis provides further intuition for the algorithm of Buchbinder et al. [FOCS'12] in the submodular case. Additionally, we show that the naive random algorithm gives a 1/4-approximation for bisubmodular functions, corresponding again to known performance guarantees for submodular functions. Thus, bisubmodular functions exhibit approximability identical to submodular functions in all of the algorithmic contexts we consider.

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“…Submodular function maximisation is easily shown to be NPhard [47] since it generalises many standard NP-hard problems such as the maximum cut problem. In contrast, the problem of minimising a submodular function (SFM) can be solved efficiently with only polynomially many oracle calls, either by using the ellipsoid algorithm [20,21], or by using one of several combinatorial algorithms that have been obtained in the last decade [46,26,24,25,40,28]. The time complexity of the fastest known general algorithm for SFM is O(n 6 + n 5 L), where n is the number of variables and L is the time required to evaluate the function [40].…”

Section: Introduction 1backgroundmentioning

confidence: 99%

The Expressive Power of Binary Submodular Functions

Živný

Cohen

Jeavons

2009

Mathematical Foundations of Computer Science 2009

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It has previously been an open problem whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This problem has been considered within several different contexts in computer science, including computer vision, artificial intelligence, and pseudo-Boolean optimisation. Using a connection between the expressive power of valued constraints and certain algebraic properties of functions, we answer this question negatively.Our results have several corollaries. First, we characterise precisely which submodular functions of arity 4 can be expressed by binary submodular functions. Next, we identify a novel class of submodular functions of arbitrary arities which can be expressed by binary submodular functions, and therefore minimised efficiently using a so-called expressibility reduction to the Min-Cut problem. More importantly, our results imply limitations on this kind of reduction and establish for the first time that it cannot be used in general to minimise arbitrary submodular functions. Finally, we refute a conjecture of Promislow and Young on the structure of the extreme rays of the cone of Boolean submodular functions.

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A Faster Scaling Algorithm for Minimizing Submodular Functions

Cited by 44 publications

References 11 publications

Submodular Function Minimization under Covering Constraints

Submodular Function Minimization under Covering Constraints

Maximizing Bisubmodular and k-Submodular Functions

The Expressive Power of Binary Submodular Functions

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