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

TL;DR

This paper studies CCMSP, a chance-constrained variant of the two-machine makespan scheduling problem with grouped stochastic processing times and covariances, using a one-sided Chebyshev surrogate to enforce probabilistic constraints. It analyzes the runtime behavior of Randomized Local Search (RLS) and the (1+1) EA on two variants CCMSP-1 and CCMSP-2, proving polynomial-time solvability for CCMSP-1 while establishing NP-hardness for CCMSP-2^+ and CCMSP-2^-; it also provides detailed runtime and approximation results for the simplified CCMSP-2 variants and validates findings with experiments. The results show that CCMSP-1 admits efficient exact solutions under both algorithms, CCMSP-2^+ is NP-hard but has tractable odd-n instances for RLS with $O( ext{poly}(n,k))$ time, and CCMSP-2^- is NP-hard with a provable 2-approximation in runtime $O(n^5)$. The experimental results corroborate the theoretical insights, illustrating the impact of stochastic elements and covariance on performance, and offering guidelines for applying EAs to chance-constrained scheduling problems in practice.

Abstract

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.

Figures (17)

• Figure 1: An illustration for the proof of Theorem \ref{['thm:RLS CCMSP-1']}.
• Figure 2: RLS for CCMSP-1 with $c = 10^{-2}$.
• Figure 3: (1+1) EA for CCMSP-1 with $c = 10^{-2}$.
• Figure 4: RLS for CCMSP-1 with $c = 10^{-7}$.
• Figure 5: (1+1) EA for CCMSP-1 with $c = 10^{-7}$.

Theorems & Definitions (26)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof
• Theorem 6