Improved Evolutionary Algorithms for Submodular Maximization with Cost Constraints

TL;DR

The paper tackles monotone submodular maximization under cost constraints (SMC), a problem where classic greedy methods achieve a $\frac{1}{2}(1-\frac{1}{e})$-approximation but can be computationally intensive. It introduces the evo-SMC evolutionary algorithm and a faster stochastic variant st-evo-SMC, both attaining a $\tfrac{1}{2}$-approximation with provable guarantees; evo-SMC runs in $\mathcal{O}(n^2K_{\beta})$ iterations while st-evo-SMC achieves $\mathcal{O}\left(\frac{nK_{\beta}\log(1/\varepsilon)}{p}\right)$ iterations with probability $1-\varepsilon$, where $p$ is a stochasticity parameter. The methods leverage a surrogate objective $g(X)=\frac{f(X)}{c(X)}$ and a mutation-based search with a good-mutation mechanism, complemented by bloom-filter accelerations to prune duplicate evaluations. Empirical evaluations on influence maximization, directed vertex cover, and sensor placement with costs show the proposed algorithms consistently outperform existing evolutionary methods and rival or exceed greedy baselines in solution quality with favorable runtimes.

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 $\mathcal{O}(n^2K_β)$ iterations, where $K_β$ denotes the maximum size of a feasible solution set with cost constraint $β$. 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 $1-ε$. The required number of iterations reduces to $\mathcal{O}(nK_β\log{(1/ε)}/p)$, where the user defined parameters $p \in (0,1]$ represents the stochasticity probability, and $ε\in (0,1]$ 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.

Figures (10)

• Figure 1: Comparisons on Film-Trust and Facebook networks.
• Figure 2: Comparisons on Eu-Email and Protein network with cost penalty $q=5$.
• Figure 3: Comparisons on Beijing data with various $\epsilon$ and $p$.
• Figure 4: At each breakpoint, we record the cost of the selected set $S$ and compute $c(S)/\beta$ in Fig. \ref{['fig:costs-app']}. We also retrieve $G_i$ and compute the number of feasible elements that can be considered to augment it.
• Figure 5: Comparisons on Film-Trust and Facebook networks with various $\epsilon$.

Theorems & Definitions (26)

• Lemma 1: Chernoff Bound
• Definition 1: Surrogate function $g(X)$
• Theorem 2
• Definition 2: Good mutation and $\omega$
• Lemma 3
• proof
• Lemma 4
• proof
• Claim 1
• proof