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A brief note on approximate optimization of submodular functions

Alen Alexanderian

TL;DR

This note tackles scalable approximate optimization of monotone submodular functions under a cardinality constraint. It first recalls the greedy algorithm, which adds at each step the element with the largest marginal gain and guarantees $f(S_k) \ge (1-1/e) \max_{S\in\mathcal{V}_k} f(S)$. It then presents two efficiency-enhancing variants: lazy greedy reduces function evaluations by exploiting submodularity with upper bounds on marginal gains, and stochastic greedy reduces evaluations further by random sampling, achieving $\mathbb{E}[f(S_k)] \ge (1-1/e-\varepsilon) \max_{S\in\mathcal{V}_k} f(S)$ with a sample size $s=\lceil( n/k) \log(1/\varepsilon)\rceil$, and can be combined with lazy evaluations. Collectively, these methods provide practical, theoretically motivated tools for large-scale submodular optimization with strong approximation guarantees.

Abstract

We briefly discuss the greedy method and a couple of its more efficient variants for approximately maximizing monotone submodular functions.

A brief note on approximate optimization of submodular functions

TL;DR

This note tackles scalable approximate optimization of monotone submodular functions under a cardinality constraint. It first recalls the greedy algorithm, which adds at each step the element with the largest marginal gain and guarantees . It then presents two efficiency-enhancing variants: lazy greedy reduces function evaluations by exploiting submodularity with upper bounds on marginal gains, and stochastic greedy reduces evaluations further by random sampling, achieving with a sample size , and can be combined with lazy evaluations. Collectively, these methods provide practical, theoretically motivated tools for large-scale submodular optimization with strong approximation guarantees.

Abstract

We briefly discuss the greedy method and a couple of its more efficient variants for approximately maximizing monotone submodular functions.
Paper Structure (6 sections, 3 theorems, 17 equations, 1 algorithm)

This paper contains 6 sections, 3 theorems, 17 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The greedy method
  4. Two variants of the greedy method
  5. Lazy greedy
  6. Stochastic greedy

Key Result

Proposition 2.3

A set function $f:{\large\text{$\mathpzc{P}$}}(V) \to \mathds{R}$ is submodular if and only if

Theorems & Definitions (8)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Example 2.4
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 4.1