[论文解读] Stochastic submodular cover with limited adaptivity

Arpit Agarwal, Sepehr Assadi+1 more|arXiv (Cornell University)|Jan 6, 2019

Complexity and Algorithms in Graphs被引用 8

一句话总结

本文提出了 r 轮自适应算法用于随机子模覆盖问题，其中决策在 r 轮中进行，每轮使用固定的物品顺序，从而在适应性受限的情况下实现近似最优性能。它证明了相对于完全自适应算法，近似比为紧致的 O(Q^{1/r})，表明对数轮次的适应性既必要也足以弥合适应性差距。

ABSTRACT

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 τk, and realizes items in the order specified by the permutation until the function value is at least τk. 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 is O(Q1/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 unbounded computational power. Thus our work shows that logarithmic rounds of adaptivity are necessary and sufficient to obtain near-optimal solutions to the stochastic submodular cover problem, and even few rounds of adaptivity are sufficient to sharply reduce the adaptivity gap.

研究动机与目标

• 为解决在随机子模覆盖问题中完全自适应算法与非自适应算法之间的巨大适应性差距。
• 研究有限适应性——特别是 r 轮——是否能够实现近似最优性能。
• 确定在随机覆盖问题中，适应性轮次数与近似质量之间的权衡关系。
• 建立 r 轮自适应算法性能的紧致上下界。

提出的方法

• 定义一种 r 轮自适应算法，即在每轮 k ∈ [r] 的开始，基于先前的实现结果，选择一个固定的排列和阈值 τk。
• 在每轮中，物品按固定顺序依次实现，直到函数值达到 τk，且排列是根据前序轮次的结果自适应选择的。
• 使用多项式时间算法计算 r 轮策略，确保期望成本在完全自适应最优解的 O(Q^{1/r}) 之内。
• 通过覆盖函数构造一个下界实例，证明即使计算能力无限，任何 r 轮算法也无法获得优于 Ω(Q^{1/r}) 的近似比。
• 利用子模性和随机物品实现的特性，对各轮的期望成本进行有界控制。
• 通过证明下界与上界在常数因子内一致，验证上界的紧致性。

实验结果

研究问题

• RQ1 r 轮自适应算法是否能在适应性受限的情况下实现随机子模覆盖问题中的近似最优性能？
• RQ2 适应性轮次数 r 与对完全自适应最优解的近似比之间存在何种权衡？
• RQ3 即使计算能力无限，r 轮算法的性能是否存在根本性限制？
• RQ4 仅通过少数几轮适应性，适应性差距是否能显著缩小？

主要发现

• 对于任意整数 r，存在一种多项式时间的 r 轮自适应算法，其期望成本为完全自适应算法的 O(Q^{1/r}) 倍。
• 上界在常数因子内是紧致的，因为存在实例使得任何 r 轮算法都无法获得优于 Ω(Q^{1/r}) 的近似比。
• 即使在覆盖函数情况下，当 r = 1 时适应性差距仍为 Ω(Q)，但随着 r 轮的引入，该差距缩小至 O(Q^{1/r})。
• 对数轮次的适应性（即 r = Θ(log Q)）既必要也足以实现对完全自适应最优解的常数因子近似。
• 即使对于计算能力无限的算法，该下界依然成立，证明该限制是结构性的，而非计算能力所致。
• 结果表明，仅需少量适应性轮次（具体为 r = O(log Q)）即可近乎完全弥合随机子模覆盖问题中的适应性差距。