Sliding Window Bi-Objective Evolutionary Algorithms for Optimizing Chance-Constrained Monotone Submodular Functions

TL;DR

This work tackles optimizing a monotone submodular function under a chance constraint by introducing Sliding Window GSEMO (SW-GSEMO), which uses a sliding-window parent-selection mechanism to cap population growth and improve runtime while retaining approximation quality. The method employs a bi-objective fitness combining the submodular gain and a surrogate weight based on Chernoff or Chebyshev bounds, enabling effective handling of stochastic weights in two settings: Uniform IID and Uniform with the same dispersion. The authors prove improved expected runtimes for SW-GSEMO (e.g., $t_{\max}=e k n\ln(nk)$ yielding a $(1-o(1))(1-1/e)$-approximation in $O(nk\log n)$ for IID weights, and a $(1/2-o(1))(1-1/e)$-approximation in $O(n((B/a_{\min})\log n + P_{\max}))$ for the dispersion setting), and validate these results empirically on the maximum coverage problem across large graphs. Experiments show SW-GSEMO often outperforms GSEMO and NSGA-II, with the surrogate choice depending on the violation probability $\alpha$, and provide visual insight into the sliding-window dynamics. Overall, the paper demonstrates scalable MOEAs for stochastic submodular optimization with practical impact on large network problems.

Abstract

Variants of the GSEMO algorithm using multi-objective formulations have been successfully analyzed and applied to optimize chance-constrained submodular functions. However, due to the effect of the increasing population size of the GSEMO algorithm considered in these studies from the algorithms, the approach becomes ineffective if the number of trade-offs obtained grows quickly during the optimization run. In this paper, we apply the sliding-selection approach introduced in [21] to the optimization of chance-constrained monotone submodular functions. We theoretically analyze the resulting SW-GSEMO algorithm which successfully limits the population size as a key factor that impacts the runtime and show that this allows it to obtain better runtime guarantees than the best ones currently known for the GSEMO. In our experimental study, we compare the performance of the SW-GSEMO to the GSEMO and NSGA-II on the maximum coverage problem under the chance constraint and show that the SW-GSEMO outperforms the other two approaches in most cases. In order to get additional insights into the optimization behavior of SW-GSEMO, we visualize the selection behavior of SW-GSEMO during its optimization process and show it beats other algorithms to obtain the highest quality of solution in variable instances.

Figures (2)

• Figure 1: Optimization process of SW-GSEMO for ca-CondaMat
• Figure 2: Different surrogate weights obtained during optimization by the SW-GSEMO based on different surrogates in ca-CSphd with IID weights

Theorems & Definitions (4)

• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• proof