Towards Minimizing k-Submodular Functions

Huber, Anna; Kolmogorov, Vladimir

doi:10.1007/978-3-642-32147-4_40

Cited by 74 publications

(79 citation statements)

References 38 publications

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“…Some initial results in this direction can be found in [24,34,35]. At the time of submission it was open whether α-bisubmodular functions can be efficiently minimized in the value-oracle model, but this question was recently answered in the positive in [20,25].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Skew Bisubmodularity and Valued CSPs

Huber¹,

Krokhin²,

Powell³

2013

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

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. An instance of the (finite-)valued constraint satisfaction problem (VCSP) is given by a finite set of variables, a finite domain of values, and a sum of (rational-valued) functions, with each function depending on a subset of the variables. The goal is to find an assignment of values to the variables that minimizes the sum. We study (assuming that PTIME = NP) how the complexity of this very general problem depends on the functions allowed in the instances. The case when the variables can take only two values was classified by Cohen et al.: essentially, submodular functions give rise to the only tractable case, and any non-submodular function can be used to express, in a certain specific sense, the NP-hard Max Cut problem. We investigate the case when the variables can take three values. We identify a new infinite family of conditions that includes bisubmodularity as a special case and which can collectively be called skew bisubmodularity. By a recent result of Thapper andŽivný, this condition implies that the corresponding VCSP can be solved by linear programming. We prove that submodularity, with respect to a total order, and skew bisubmodularity give rise to the only tractable cases, and, in all other cases, again, Max Cut can be expressed. We also show that our characterization of tractable cases is tight; that is, none of the conditions can be omitted.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Skew Bisubmodularity and Valued CSPs

Huber¹,

Krokhin²,

Powell³

2013

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

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“…Our quest of extending combinatorial algorithms for submodular functions is motivated by questions about the tractability of submodular function minimization defined on general discrete structures such as semilattices and sets of transversals (see, e.g., [10,11,13,14,[18][19][20]), where bisubmodular functions are special cases of submodular functions on a semilattice [13]. New techniques presented here might also be useful for other classes of submodular functions.…”

Section: Given a Partitionmentioning

confidence: 99%

“…The main body of the algorithm will also be used in the strongly polynomial time algorithm given in the next section, and hence we shall refer to it as REFINE. An iteration of the while-loop in REFINE (i.e., lines [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] corresponds to a scaling phase with a scaling parameter δ discussed above. …”

Section: Lemma 7 Let W Be the Set Of Vertices In G(ψ) Reachable From Smentioning

confidence: 99%

Polynomial combinatorial algorithms for skew-bisubmodular function minimization

Fujishige

Tanigawa

2017

Math. Program.

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Huber et al. (SIAM J Comput 43:1064-1084) introduced a concept of skew bisubmodularity, as a generalization of bisubmodularity, in their complexity dichotomy theorem for valued constraint satisfaction problems over the three-value domain, and Huber and Krokhin (SIAM J Discrete Math 28:1828-1837, 2014) showed the oracle tractability of minimization of skew-bisubmodular functions. Fujishige et al.(Discrete Optim 12:1-9, 2014) also showed a min-max theorem that characterizes the skew-bisubmodular function minimization, but devising a combinatorial polynomial algorithm for skew-bisubmodular function minimization was left open. In the present paper we give first combinatorial (weakly and strongly) polynomial algorithms for skew-bisubmodular function minimization.

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“…Related work The name of k-submodular functions was first introduced in [15] but the concept has been known since at least [7]. k-submodularity is a special case of strong tree submodularity [23] with the tree being a star on k + 1 vertices.…”

Section: Introductionmentioning

confidence: 99%

“…To the best of our knowledge, it is not known whether the ellipsoid method can be employed for minimizing k-submodular functions for k ≥ 3 (some partial results can be found in [15]), let alone whether there is a (fully) combinatorial algorithm for minimizing k-submodular functions for k ≥ 3. However, it has recently been shown that explicitly given k-submodular functions can be minimized in polynomial time [36].…”

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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Towards Minimizing k-Submodular Functions

Cited by 74 publications

References 38 publications

Skew Bisubmodularity and Valued CSPs

Skew Bisubmodularity and Valued CSPs

Polynomial combinatorial algorithms for skew-bisubmodular function minimization

Maximizing Bisubmodular and k-Submodular Functions

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