Maximizing k -Submodular Functions and Beyond

Abstract

We consider the maximization problem in the value oracle model of functions defined on k-tuples of sets that are submodular in every orthant and r-wise monotone, where k ≥ 2 and 1 ≤ r ≤ k. We give an analysis of a deterministic greedy algorithm that shows that any such function can be approximated to a factor of 1/(1 + r). For r = k, we give an analysis of a randomised greedy algorithm that shows that any such function can be approximated to a factor of 1/(1 + k/2).In the case of k = r = 2, the considered functions correspond precisely to bisubmodular functions, in which case we obtain 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 .Extending a result of Ando et al., we show that for any k ≥ 3 submodularity in every orthant and pairwise monotonicity (i.e. r = 2) precisely characterize k-submodular functions. Consequently, we obtain an approximation guarantee of 1/3 (and thus independent of k) for the maximization problem of k-submodular functions.

Improved Approximation Algorithms for k-Submodular Function Maximization

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This paper presents a polynomial-time 1/2-approximation algorithm for maximizing nonnegative k-submodular functions. This improves upon the previous max{1/3, 1/(1+a)}-approximation by Ward andŽivný [15], where a = max{1, (k − 1)/4}. We also show that for monotone ksubmodular functions there is a polynomial-time k/(2k − 1)-approximation algorithm while for any ε > 0 a ((k + 1)/2k + ε)-approximation algorithm for maximizing monotone k-submodular functions would require exponentially many queries.

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“…Ward andŽivný [15] and the present authors [11] independently observed that algorithms for submodular function maximization due to Buchbinder, Feldman, Naor, and Schwartz [2] can be naturally extended to bisubmodular function maximization. In particular the randomized double greedy algorithm for submodular functions can be seen as a randomized greedy algorithm in the bisubmodular setting and it achieves the best approximation ratio 1/2.…”

“…In particular the randomized double greedy algorithm for submodular functions can be seen as a randomized greedy algorithm in the bisubmodular setting and it achieves the best approximation ratio 1/2. Ward andŽivný [15] further analyzed the randomized greedy algorithm for k-submodular function maximization and proved that its approximation ratio is 1/(1 + a), where a = max{1, (k − 1)/4}. They also gave a deterministic 1/3-approximation algorithm.…”

On $k$-Submodular Relaxation

“…We here consider maximization of functions having no +∞. Following [7,13], just recently, Iwata, Tanigawa, and Yoshida [8] presented a 1/2-approximation algorithm for nonnegative k-submodular function maximization. ‡ Note that our definition of k-submodular relaxation is slightly different from the one given by [9], where the definition in [9] requires one more condition min g = min f .…”

Derandomization for k-Submodular Maximization

“…We also give randomized √ 17−3 2 approximation algorithm for k = 3. We use the same framework used in [11] and [15] with different probabilities.…”