Stability and Recovery for Independence Systems

TL;DR

This paper investigates the stability of greedy and local search algorithms for maximizing monotone submodular functions under feasibility constraints, identifying thresholds for guaranteed optimal recovery and introducing stability concepts for non-additive functions.

Contribution

It introduces the first stability framework for non-additive functions and determines the stability threshold for recovery guarantees in submodular maximization.

Findings

Identifies stability thresholds for greedy and local search algorithms.

Defines perturbation-stability for non-additive functions.

Resolves the approximation guarantee of local search in p-extendible systems.

Abstract

Two genres of heuristics that are frequently reported to perform much better on "real-world" instances than in the worst case are greedy algorithms and local search algorithms. In this paper, we systematically study these two types of algorithms for the problem of maximizing a monotone submodular set function subject to downward-closed feasibility constraints. We consider perturbation-stable instances, in the sense of Bilu and Linial, and precisely identify the stability threshold beyond which these algorithms are guaranteed to recover the optimal solution. Byproducts of our work include the first definition of perturbation-stability for non-additive objective functions, and a resolution of the worst-case approximation guarantee of local search in p-extendible systems.