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Maximization of Approximately Submodular Functions

Thibaut Horel, Yaron Singer

TL;DR

The query-complexity of maximizing F, a function that is approximately submodular under a cardinality constraint, is characterized as a function of the error level $\eps > 0$ and both lower and upper bounds are provided.

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.

Maximization of Approximately Submodular Functions

TL;DR

The query-complexity of maximizing F, a function that is approximately submodular under a cardinality constraint, is characterized as a function of the error level and both lower and upper bounds are provided.

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 is -approximately submodular if there exists a submodular function such that for all subsets . We are interested in characterizing the query-complexity of maximizing subject to a cardinality constraint as a function of the error level . We provide both lower and upper bounds: for we show an exponential query-complexity lower bound. In contrast, when or under a stronger bounded curvature assumption, we give constant approximation algorithms.

Paper Structure

This paper contains 27 sections, 9 theorems, 40 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Approximate submodularity.
  3. Optimization of approximate submodularity.
  4. Overview of the results
  5. Approximation guarantees.
  6. Discussion and additional related work
  7. Additive vs multiplicative approximation.
  8. Error vs noise.
  9. Implications of results in this paper.
  10. Query-complexity lower bounds
  11. A general overview of query-complexity lower bounds.
  12. Monotone submodular functions
  13. Constructing a class of hard functions.
  14. Indistinguishability.
  15. Case 1:
  16. ...and 12 more sections

Key Result

Lemma 1

Let $H\subseteq N$ be a set drawn uniformly among sets of size $h$, then for any $S\subseteq N$, writing $\mu = \frac{|S|h}{n}$, for any $\varepsilon$ such that $\varepsilon^2\mu>1$:

Theorems & Definitions (20)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof
  • Proposition 6
  • Theorem 7
  • ...and 10 more