[Paper Review] Maximization of Approximately Submodular Functions

Thibaut Horel, Yaron Singer|arXiv (Cornell University)|Nov 17, 2024

Complexity and Algorithms in Graphs25 references65 citations

TL;DR

The paper studies maximizing epsilon-approximately submodular functions under a cardinality constraint, providing lower bounds on query complexity and identifying conditions for constant-factor approximations.

ABSTRACT

We study the problem of maximizing a function that is approximately submodular under a cardinality constraint. Approximate submodularity implicitly appears in a wide range of applications as in many cases errors in evaluation of a submodular function break submodularity. Say that $F$ is $\varepsilon$-approximately submodular if there exists a submodular function $f$ such that $(1-\varepsilon)f(S) \leq F(S)\leq (1+\varepsilon)f(S)$ for all subsets $S$. We are interested in characterizing the query-complexity of maximizing $F$ subject to a cardinality constraint $k$ as a function of the error level $\varepsilon>0$. We provide both lower and upper bounds: for $\varepsilon>n^{-1/2}$ we show an exponential query-complexity lower bound. In contrast, when $\varepsilon< {1}/{k}$ or under a stronger bounded curvature assumption, we give constant approximation algorithms.

Motivation & Objective

• Motivate and formalize the problem of maximizing an epsilon-approximately submodular function under a cardinality constraint.
• Characterize query complexity versus error level ε for general and specialized function classes.
• Identify conditions under which constant-factor approximation is possible (and tight) for approximate submodularity.

Proposed method

• Define epsilon-approximately submodular functions F via a submodular representative f with (1-ε)f(S) ≤ F(S) ≤ (1+ε)f(S) for all S.
• Derive exponential query-complexity lower bounds for general monotone submodular functions when ε ≥ n^{-1/2} (Theorem 3) and for coverage functions when ε ≥ n^{-1/3} (Theorem 4).
• Prove positive results: greedy achieves (1-1/e - O(δ)) when ε ≤ δ/k (Theorem 5), and under bounded curvature c, an algorithm achieves (1-c)( (1-ε)/(1+ε) )^2.
• Show tightness: greedy with ε = 1/k does not guarantee constant-factor approximation (Proposition 6).
• Extend results to matroid constraints and discuss implications for noise models and PMAC learning.

Experimental results

Research questions

• RQ1 What is the query complexity to maximize an ε-approximately submodular function under a cardinality constraint?
• RQ2 Do constant-factor approximations exist for small ε and/or under additional structure such as bounded curvature?
• RQ3 How do lower and upper bounds differ between general monotone submodular and structured classes like coverage functions?
• RQ4 What is the impact of ε relative to k on the performance of greedy-like algorithms?
• RQ5 Can results extend to matroid constraints beyond cardinality constraints?

Key findings

• Exponential query complexity lower bounds hold for ε ≥ n^{-1/2} in the general monotone submodular case.
• Exponential query complexity lower bounds also hold for coverage functions with ε ≥ n^{-1/3}.
• Greedy yields a constant-factor approximation when ε ≤ δ/k (for any fixed δ ∈ (0,1)) with a ratio approaching 1-1/e as δ→0.
• Under bounded curvature c, there exists an algorithm achieving a constant approximation (1-c)((1-ε)/(1+ε))^2 for any ε.
• The ε = 1/k threshold is tight for the greedy algorithm in achieving constant-factor guarantees (non-constant performance for larger ε).
• Results extend to matroid constraints with analogous constants (greedy guarantees reduce by factors).