Improved Evolutionary Algorithms for Submodular Maximization with Cost Constraints

TL;DR

This paper introduces improved evolutionary algorithms for submodular maximization with cost constraints, achieving the best known approximation guarantees and demonstrating superior empirical performance over existing methods.

Contribution

The paper develops evo-SMC and its stochastic variant st-evo-SMC, providing the first high-probability approximation guarantees for evolutionary algorithms tackling this problem.

Findings

evo-SMC achieves 1/2-approximation with high probability within O(n^2K_β) iterations

st-evo-SMC reduces iteration complexity while maintaining approximation ratio

Empirical results show the proposed algorithms outperform existing methods in solution quality

Abstract

We present an evolutionary algorithm evo-SMC for the problem of Submodular Maximization under Cost constraints (SMC). Our algorithm achieves $1/2$-approximation with a high probability $1-1/n$ within $\mathcal{O}(n^2K_{\beta})$ iterations, where $K_{\beta}$ denotes the maximum size of a feasible solution set with cost constraint $\beta$. To the best of our knowledge, this is the best approximation guarantee offered by evolutionary algorithms for this problem. We further refine evo-SMC, and develop st-evo-SMC. This stochastic version yields a significantly faster algorithm while maintaining the approximation ratio of $1/2$, with probability $1-\epsilon$. The required number of iterations reduces to $\mathcal{O}(nK_{\beta}\log{(1/\epsilon)}/p)$, where the user defined parameters $p \in (0,1]$ represents the stochasticity probability, and $\epsilon \in (0,1]$ denotes the error threshold. Finally, the empirical evaluations carried out through extensive experimentation substantiate the efficiency and effectiveness of our proposed algorithms. Our algorithms consistently outperform existing methods, producing higher-quality solutions.