Discrete Stochastic Submodular Maximization: Adaptive vs. Non-adaptive vs. Offline

Hellerstein, Lisa; Kletenik, Devorah; Lin, Patrick

doi:10.1007/978-3-319-18173-8_17

Cited by 7 publications

(6 citation statements)

References 11 publications

(23 reference statements)

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“…It was shown that there is a non-adaptive algorithm that achieves an approximation ratio of 1/3 with respect to an optimal adaptive algorithm. In [31], the problem of stochastic submodular maximization was also studied under various types of constraints, including knapsack constraints. An approximation ratio of τ for this problem under knapsack constraint was given, where τ is the smallest probability of any element in the ground set being realized by any item.…”

Section: Related Workmentioning

confidence: 99%

Stochastic Submodular Cover with Limited Adaptivity

Agarwal¹,

Assadi²,

Khanna³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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In the submodular cover problem, we are given a non-negative monotone submodular function f over a ground set E of items, and the goal is to choose a smallest subset S ⊆ E such that f (S) = Q where Q = f (E). In the stochastic version of the problem, we are given m stochastic items which are different random variables that independently realize to some item in E, and the goal is to find a smallest set of stochastic items whose realization R satisfies f (R) = Q. The problem captures as a special case the stochastic set cover problem and more generally, stochastic covering integer programs.A fully adaptive algorithm for stochastic submodular cover chooses an item to realize and based on its realization, decides which item to realize next. A non-adaptive algorithm on the other hand needs to choose a permutation of items beforehand and realize them one by one in the order specified by this permutation until the function value reaches Q. The cost of the algorithm in both case is the number (or costs) of items realized by the algorithm. It is not difficult to show that even for the coverage function there exist instances where the expected cost of a fully adaptive algorithm and a non-adaptive algorithm are separated by Ω(Q). This strong separation, often referred to as the adaptivity gap, is in sharp contrast to the separations observed in the framework of stochastic packing problems where the performance gap for many natural problem is close to the poly-time approximability of the non-stochastic version of the problem. Motivated by this striking gap between the power of adaptive and non-adaptive algorithms, we consider the following question in this work: does one need full power of adaptivity to obtain a near-optimal solution to stochastic submodular cover? In particular, how does the performance guarantees change when an algorithm interpolates between these two extremes using a few rounds of adaptivity.Towards this end, we define an r-round adaptive algorithm to be an algorithm that chooses a permutation of all available items in each round k ∈ [r], and a threshold τ k , and realizes items in the order specified by the permutation until the function value is at least τ k . The permutation for each round k is chosen adaptively based on the realization in the previous rounds, but the ordering inside each round remains fixed regardless of the realizations seen inside the round. Our main result is that for any integer r, there exists a poly-time r-round adaptive algorithm for stochastic submodular cover whose expected cost isÕ(Q 1/r ) times the expected cost of a fully adaptive algorithm. Prior to our work, such a result was not known even for the case of r = 1 and when f is the coverage function. On the other hand, we show that for any r, there exist instances of the stochastic submodular cover problem where no r-round adaptive algorithm can achieve better than Ω(Q 1/r ) approximation to the expected cost of a fully adaptive algorithm. Our lower bound result holds even for coverage function and for algorithms with unbo...

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Section: Related Workmentioning

confidence: 99%

Stochastic Submodular Cover with Limited Adaptivity

Agarwal¹,

Assadi²,

Khanna³

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…All except the orienteering results rely on having relaxations that capture the constraints of the problem via linear constraints. For stochastic monotone submodular functions where the probing constraints are given by matroids, Asadpour et al [AN16] bounded the adaptivity gap by e e´1 ; Hellerstein et al [HKL15] bound it by 1 τ , where τ is the smallest probability of some set being materialized. Other relevant papers are [LPRY08,DHK14].…”

Section: Further Related Workmentioning

confidence: 99%

(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing

Domagoj¹,

Singla²,

Zuzic³

2019

Preprint

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Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge e exists, we need to perform an expensive test, in which case the edge e appears independently with a known probability p e . Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size.The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is adaptive-decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the adaptivity gap is small, i.e., the best non-adaptive strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values.Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most 2 and provide a matching lower bound. For a weighted rank function of a k-extendible system (a generalization of intersection of k matroids), we show the adaptivity gap is between Opk log kq and k. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al. [GNS17].

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“…For stochastic monotone submodular functions where the probing constraints are given by matroids, Asadpour et al [4] bounded the adaptivity gap by e e−1 ; Hellerstein et al [32] bound it by 1 τ , where τ is the smallest probability of some set being materialized. (See also [37,19].…”

Section: Related Workmentioning

confidence: 99%

Adaptivity Gaps for Stochastic Probing: Submodular and XOS Functions

Gupta¹,

Nagarajan²,

Singla³

2017

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

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Suppose we are given a submodular function f over a set of elements, and we want to maximize its value subject to certain constraints. Good approximation algorithms are known for such problems under both monotone and non-monotone submodular functions. We consider these problems in a stochastic setting, where elements are not all active and we only get value from active elements. Each element e is active independently with some known probability p e , but we don't know the element's status a priori : we find it out only when we probe the element e. Moreover, the sequence of elements we probe must satisfy a given prefix-closed constraint, e.g., matroid, orienteering, deadline, precedence, or any downward-closed constraint.In this paper we study the gap between adaptive and non-adaptive strategies for f being a submodular or a fractionally subadditive (XOS) function. If this gap is small, we can focus on finding good non-adaptive strategies instead, which are easier to find as well as to represent. We show that the adaptivity gap is a constant for monotone and non-monotone submodular functions, and logarithmic for XOS functions of small width. These bounds are nearly tight. Our techniques show new ways of arguing about the optimal adaptive decision tree for stochastic optimization problems.

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Discrete Stochastic Submodular Maximization: Adaptive vs. Non-adaptive vs. Offline

Cited by 7 publications

References 11 publications

Stochastic Submodular Cover with Limited Adaptivity

Stochastic Submodular Cover with Limited Adaptivity

(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing

Adaptivity Gaps for Stochastic Probing: Submodular and XOS Functions

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