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Archive-based Single-Objective Evolutionary Algorithms for Submodular Optimization

Frank Neumann, Günter Rudolph

TL;DR

The paper investigates constrained submodular maximization, a class of NP-hard problems, and asks whether simple single-objective EAs can yield provable performance. It introduces two variants: a (1+λ)-EA without archive for uniform constraints and a archive-based (1+1)-EA for general monotone costs, both designed to progressively expand feasibility. The authors prove a $(1-1/e)$-approximation for the uniform case with a runtime bound $t_{max}=2 e r n \log(n)$ and a $(\alpha_f/2)(1- e^{-\alpha_f})$-approximation for general costs, each holding with high probability. Experiments on maximum-coverage problems show these single-objective methods are competitive with, and often outperform, multiobjective GSEMO-based approaches, especially on larger instances. Overall, the work demonstrates that carefully designed single-objective EAs can match the performance of more complex multiobjective methods for constrained submodular optimization, with meaningful implications for practice.

Abstract

Constrained submodular optimization problems play a key role in the area of combinatorial optimization as they capture many NP-hard optimization problems. So far, Pareto optimization approaches using multi-objective formulations have been shown to be successful to tackle these problems while single-objective formulations lead to difficulties for algorithms such as the $(1+1)$-EA due to the presence of local optima. We introduce for the first time single-objective algorithms that are provably successful for different classes of constrained submodular maximization problems. Our algorithms are variants of the $(1+λ)$-EA and $(1+1)$-EA and increase the feasible region of the search space incrementally in order to deal with the considered submodular problems.

Archive-based Single-Objective Evolutionary Algorithms for Submodular Optimization

TL;DR

The paper investigates constrained submodular maximization, a class of NP-hard problems, and asks whether simple single-objective EAs can yield provable performance. It introduces two variants: a (1+λ)-EA without archive for uniform constraints and a archive-based (1+1)-EA for general monotone costs, both designed to progressively expand feasibility. The authors prove a -approximation for the uniform case with a runtime bound and a -approximation for general costs, each holding with high probability. Experiments on maximum-coverage problems show these single-objective methods are competitive with, and often outperform, multiobjective GSEMO-based approaches, especially on larger instances. Overall, the work demonstrates that carefully designed single-objective EAs can match the performance of more complex multiobjective methods for constrained submodular optimization, with meaningful implications for practice.

Abstract

Constrained submodular optimization problems play a key role in the area of combinatorial optimization as they capture many NP-hard optimization problems. So far, Pareto optimization approaches using multi-objective formulations have been shown to be successful to tackle these problems while single-objective formulations lead to difficulties for algorithms such as the -EA due to the presence of local optima. We introduce for the first time single-objective algorithms that are provably successful for different classes of constrained submodular maximization problems. Our algorithms are variants of the -EA and -EA and increase the feasible region of the search space incrementally in order to deal with the considered submodular problems.
Paper Structure (13 sections, 5 theorems, 20 equations, 1 figure, 2 tables, 2 algorithms)

This paper contains 13 sections, 5 theorems, 20 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $(1+\lambda)$-EA without Archive
  4. Algorithm
  5. Uniform Constraint
  6. General Constraint
  7. $(1+1)$-EA with Archive
  8. Algorithm
  9. Analysis
  10. Case 1:
  11. Case 2:
  12. Experimental Investigations
  13. Conclusions

Key Result

Theorem 1

The following conditions are equivalent to the definition of submodular set functions: a) for all $A\subseteq B\subseteq U$ and $x\notin B$ b) for all $A\subseteq B\subseteq U$

Figures (1)

  • Figure 1: Gray set $A$ in the left figure is a subset of the gray set $B$ in the middle figure which in turn is a subset of the gray set $C$ in the right figure. Adding the blue set leads to a lower gain of area the larger the gray set is.

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1: see DBLP:journals/mp/NemhauserWF78, Proposition 2.1
  • Lemma 1
  • proof
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 2 more