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

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.

Sampling-based Pareto Optimization for Chance-constrained Monotone Submodular Problems

Recently surrogate functions based on the tail inequalities were developed to evaluate the chance constraints in the context of evolutionary computation and several Pareto optimization algorithms using these surrogates were successfully applied in optimizing chance-constrained monotone submodular problems. However, the difference in performance between algorithms using the surrogates and those employing the direct sampling-based evaluation remains unclear. Within the paper, a sampling-based method is proposed to directly evaluate the chance constraint. Furthermore, to address the problems with more challenging settings, an enhanced GSEMO algorithm integrated with an adaptive sliding window, called ASW-GSEMO, is introduced. In the experiments, the ASW-GSEMO employing the sampling-based approach is tested on the chance-constrained version of the maximum coverage problem with different settings. Its results are compared with those from other algorithms using different surrogate functions. The experimental findings indicate that the ASW-GSEMO with the sampling-based evaluation approach outperforms other algorithms, highlighting that the performances of algorithms using different evaluation methods are comparable. Additionally, the behaviors of ASW-GSEMO are visualized to explain the distinctions between it and the algorithms utilizing the surrogate functions.

Optimizing Chance-Constrained Submodular Problems with Variable Uncertainties

Chance constraints are frequently used to limit the probability of constraint violations in real-world optimization problems where the constraints involve stochastic components. We study chance-constrained submodular optimization problems, which capture a wide range of optimization problems with stochastic constraints. Previous studies considered submodular problems with stochastic knapsack constraints in the case where uncertainties are the same for each item that can be selected. However, uncertainty levels are usually variable with respect to the different stochastic components in real-world scenarios, and rigorous analysis for this setting is missing in the context of submodular optimization. This paper provides the first such analysis for this case, where the weights of items have the same expectation but different dispersion. We present greedy algorithms that can obtain a high-quality solution, i.e., a constant approximation ratio to the given optimal solution from the deterministic setting. In the experiments, we demonstrate that the algorithms perform effectively on several chance-constrained instances of the maximum coverage problem and the influence maximization problem.

Runtime Performance of Evolutionary Algorithms for the Chance-constrained Makespan Scheduling Problem

The Makespan Scheduling problem is an extensively studied NP-hard problem, and its simplest version looks for an allocation approach for a set of jobs with deterministic processing times to two identical machines such that the makespan is minimized. However, in real life scenarios, the actual processing time of each job may be stochastic around the expected value with a variance, under the influence of external factors, and the actual processing times of these jobs may be correlated with covariances. Thus within this paper, we propose a chance-constrained version of the Makespan Scheduling problem and investigate the theoretical performance of the classical Randomized Local Search and (1+1) EA for it. More specifically, we first study two variants of the Chance-constrained Makespan Scheduling problem and their computational complexities, then separately analyze the expected runtime of the two algorithms to obtain an optimal solution or almost optimal solution to the instances of the two variants. In addition, we investigate the experimental performance of the two algorithms for the two variants.