[Paper Review] Adaptivity gaps for stochastic probing: submodular and XOS functions

Anupam Gupta, Viswanath Nagarajan+1 more|arXiv (Cornell University)|Jan 16, 2017

Auction Theory and Applications35 citations

TL;DR

This paper studies the adaptivity gap in stochastic probing for submodular and XOS (fractionally subadditive) functions, where elements are randomly active and must be probed to reveal their status. It establishes that the gap is a constant for monotone and non-monotone submodular functions and logarithmic for XOS functions of small width, showing that non-adaptive strategies can be nearly as effective as adaptive ones in these cases.

ABSTRACT

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 pe, 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.

Motivation & Objective

• To understand the trade-off between adaptive and non-adaptive strategies in stochastic probing under submodular and XOS objective functions.
• To quantify the adaptivity gap—the ratio between optimal adaptive and non-adaptive strategies—for various function classes and constraints.
• To develop new analytical techniques for reasoning about optimal adaptive decision trees in stochastic optimization.
• To establish nearly tight bounds on the adaptivity gap for monotone and non-monotone submodular functions, and for XOS functions of small width.
• To demonstrate that non-adaptive strategies can be nearly optimal, simplifying both computation and representation in practice.

Proposed method

• Analyzes the adaptivity gap using a novel framework for comparing adaptive and non-adaptive strategies in stochastic probing.
• Applies structural properties of submodular and XOS functions to bound the performance difference between adaptive and non-adaptive policies.
• Uses a recursive decomposition technique to reason about optimal decision trees in the stochastic setting.
• Leverages known results on submodular maximization under constraints (e.g., matroid, orienteering) to derive bounds on the adaptivity gap.
• Introduces a width parameter for XOS functions to refine the adaptivity gap bound, showing logarithmic dependence on this width.
• Employs probabilistic analysis and concentration arguments to compare expected values of adaptive and non-adaptive policies.

Experimental results

Research questions

• RQ1 What is the maximum possible adaptivity gap for monotone submodular functions under stochastic probing with prefix-closed constraints?
• RQ2 How does the adaptivity gap behave for non-monotone submodular functions in the same stochastic setting?
• RQ3 What is the adaptivity gap for XOS functions, particularly when the function has small width?
• RQ4 Can non-adaptive strategies achieve a constant or logarithmic approximation to the optimal adaptive strategy in these settings?
• RQ5 What new analytical techniques can be developed to reason about optimal adaptive decision trees in stochastic optimization?

Key findings

• The adaptivity gap is bounded by a constant for both monotone and non-monotone submodular functions under stochastic probing with prefix-closed constraints.
• For XOS functions of small width, the adaptivity gap is logarithmic in the width, and this bound is nearly tight.
• The results imply that non-adaptive strategies can be nearly optimal for submodular and low-width XOS functions, simplifying algorithm design.
• The analysis introduces new techniques for reasoning about optimal adaptive decision trees in stochastic settings, which may generalize beyond the current scope.
• The bounds on the adaptivity gap are nearly tight, indicating that the derived ratios cannot be significantly improved in general.
• The framework applies to a wide range of constraints, including matroid, orienteering, deadline, precedence, and downward-closed constraints.