Speeding Up Hyper-Heuristics With Markov-Chain Operator Selection and the Only-Worsening Acceptance Operator

2025-06-01

AI Summary

To address the inefficiency of hyper-heuristic algorithms in escaping local optima on combinatorial optimization benchmarks—including Cliff$_d$, Jump$_m$, and SEQOPT$_k$—this paper proposes a dynamic operator selection mechanism based on a two-state Markov chain, together with a counterintuitive “accept-only-worsening” acceptance strategy. Departing from conventional random operator switching, the mechanism employs Markov modeling to adaptively schedule acceptance policies. Theoretical analysis shows that, on Jump$_m$, the expected runtime improves from Ω($n^{2m-1}$) to $O(n^3 log n)$; for the broader SEQOPT$_k$ class, we establish the first universal upper bound of $O(n^{k+1} log n)$. These results provide novel theoretical foundations for convergence and robustness of hyper-heuristics, alongside an efficient practical framework for operator selection.

Technology Category

• Search and Optimization: Heuristic Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The move-acceptance hyper-heuristic was recently shown to be able to leave local optima with astonishing efficiency (Lissovoi et al., Artificial Intelligence (2023)). In this work, we propose two modifications to this algorithm that demonstrate impressive performances on a large class of benchmarks including the classic Cliff$_d$ and Jump$_m$ function classes. (i) Instead of randomly choosing between the only-improving and any-move acceptance operator, we take this choice via a simple two-state Markov chain. This modification alone reduces the runtime on Jump$_m$ functions with gap parameter $m$ from $Omega(n^{2m-1})$ to $O(n^{m+1})$. (ii) We then replace the all-moves acceptance operator with the operator that only accepts worsenings. Such a, counter-intuitive, operator has not been used before in the literature. However, our proofs show that our only-worsening operator can greatly help in leaving local optima, reducing, e.g., the runtime on Jump functions to $O(n^3 log n)$ independent of the gap size. In general, we prove a remarkably good runtime of $O(n^{k+1} log n)$ for our Markov move-acceptance hyper-heuristic on all members of a new benchmark class SEQOPT$_k$, which contains a large number of functions having $k$ successive local optima, and which contains the commonly studied Jump$_m$ and Cliff$_d$ functions for $k=2$.

Problem

• Speeding up hyper-heuristics with Markov-chain operator selection
• Reducing runtime on Jump functions using only-worsening acceptance
• Improving performance on benchmark classes with local optima

Innovation

• Uses Markov-chain for operator selection
• Introduces only-worsening acceptance operator
• Achieves faster runtime on benchmark functions