Budget-Smoothed Analysis for Submodular Maximization

TL;DR

This paper introduces a budget smoothed analysis framework for submodular maximization, showing that greedy algorithms perform better under realistic budget distributions and characterizing the robustness of hard functions.

Contribution

It develops a new smoothed analysis approach for budgets in submodular maximization, proving greedy's optimality and characterizing worst-case functions under this framework.

Findings

Greedy is optimal for all budget distributions.

Under realistic budgets, greedy achieves better approximation guarantees.

Existence of hard functions robust to budget variations.

Abstract

The greedy algorithm for monotone submodular function maximization subject to cardinality constraint is guaranteed to approximate the optimal solution to within a $1-1/e$ factor. Although it is well known that this guarantee is essentially tight in the worst case -- for greedy and in fact any efficient algorithm, experiments show that greedy performs better in practice. We observe that for many applications in practice, the empirical distribution of the budgets (i.e., cardinality constraints) is supported on a wide range, and moreover, all the existing hardness results in theory break under a large perturbation of the budget. To understand the effect of the budget from both algorithmic and hardness perspectives, we introduce a new notion of budget smoothed analysis. We prove that greedy is optimal for every budget distribution, and we give a characterization for the worst-case submodular functions. Based on these results, we show that on the algorithmic side, under realistic budget distributions, greedy and related algorithms enjoy provably better approximation guarantees, that hold even for worst-case functions, and on the hardness side, there exist hard functions that are fairly robust to all the budget distributions.