On $k$-Submodular Relaxation

Hirai, Hiroshi; Iwamasa, Yuni

doi:10.1137/15m101926x

Cited by 16 publications

(5 citation statements)

References 12 publications

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“…, k} n defined by submodular-type inequalities. This generalization of (bi)submodular functions has recently gained attention for algorithm design and modeling [9,11,12,17,18].…”

Section: Introductionmentioning

confidence: 99%

A compact representation for minimizers of k-submodular functions

Hirai

Oki

2017

J Comb Optim

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A k-submodular function is a generalization of submodular and bisubmodular functions. This paper establishes a compact representation for minimizers of a k-submodular function by a poset with inconsistent pairs (PIP). This is a generalization of Ando-Fujishige's signed poset representation for minimizers of a bisubmodular function. We completely characterize the class of PIPs (elementary PIPs) arising from k-submodular functions. We give algorithms to construct the elementary PIP of minimizers of a k-submodular function f for three cases: (i) a minimizing oracle of f is available, (ii) f is network-representable, and (iii) f arises from a Potts energy function. Furthermore, we provide an efficient enumeration algorithm for all maximal minimizers of a Potts k-submodular function. Our results are applicable to obtain all maximal persistent labelings in actual computer vision problems. We present experimental results for real vision instances.

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“…, k} n defined by submodular-type inequalities. This generalization of (bi)submodular functions has recently gained attention for algorithm design and modeling [9,11,12,17,18].…”

Section: Introductionmentioning

confidence: 99%

A compact representation for minimizers of k-submodular functions

Hirai

Oki

2017

J Comb Optim

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“…12) we modify D 1 (x) toD 1 (x) = D 1 (x) \ {d • , d • } ∪ {d ↑ , d ↓ }, (8.13) where the incomparable pair {d • , d • } is replaced by a comparable pair {d ↑ , d ↓ } with d ↑ ≥ d ↓ . Lemma 8.7.…”

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confidence: 99%

Discrete Midpoint Convexity

et al. 2020

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For a function defined on the integer lattice, we consider discrete versions of midpoint convexity, which offer a unifying framework for discrete convexity of functions, including integral convexity, L -convexity, and submodularity. By considering discrete midpoint convexity for all pairs at ∞ -distance equal to two or not smaller than two, we identify new classes of discrete convex functions, called locally and globally discrete midpoint convex functions. These functions enjoy nice structural properties. They are stable under scaling and addition, and satisfy a family of inequalities named parallelogram inequalities. Furthermore, they admit a proximity theorem with the same small proximity bound as that for L -convex functions. These structural properties allow us to develop an algorithm for the minimization of locally and globally discrete midpoint convex functions based on the proximity-scaling approach and on a novel 2-neighborhood steepest descent algorithm.

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“…One decade ago, k-submodular functions were introduced by Huber and Kolmogorov [5] to express submodularity on choosing k disjoint sets of elements instead of a single set. Subsequently, ksubmodular functions have become a popular research direction [3,4,8,14,21], especially the problem of maximizing k-submodular functions.…”

Section: Related Workmentioning

confidence: 99%

Improved Analysis of Greedy Algorithm on k-Submodular Knapsack

Tang,

Wang

2023

Frontiers in Artificial Intelligence and Applications

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A k-submodular function is a generalization of submodular functions that takes k disjoint subsets as input and outputs a real value. It captures many problems in combinatorial optimization and machine leaning such as influence maximization, sensor placement, feature selection, etc. In this paper, we consider the monotone k-submodular maximization problem under a knapsack constraint, and explore the performance guarantee of a greedy-based algorithm: enumerating all size-2 solutions and extending every singleton solution greedily; the best outcome is returned. We provide a novel analysis framework and prove that this algorithm achieves an approximation ratio of at least 0.328. This is the best-known result of combinatorial algorithms on k-submodular knapsack maximization. In addition, within the framework, we can further improve the approximation ratio to a value approaching 1/3 with any desirable accuracy, by enumerating sufficiently large base solutions. The results can even be extended to non-monotone k-submodular functions.

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On $k$-Submodular Relaxation

Cited by 16 publications

References 12 publications

A compact representation for minimizers of k-submodular functions

A compact representation for minimizers of k-submodular functions

Discrete Midpoint Convexity

Improved Analysis of Greedy Algorithm on k-Submodular Knapsack

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