To Repair or Not to Repair? Investigating the Importance of AB-Cycles for the State-of-the-Art TSP Heuristic EAX

TL;DR

This work addresses the performance gap in the Stage I component of the state-of-the-art TSP solver EAX by deriving a necessary-and-sufficient fast check for AB-cycle validity based on $C$-vertices, enabling faster or repair-free offspring generation. It introduces two EAX variants (only-complete and ratio-based) and fixes the Stage II offspring count to 20, then benchmarks them across 10,000 evolved instances. The ratio-based variant substantially reduces PAR10 on several instance classes, particularly small LKH-friendly ones, while overall performance remains competitive across others, highlighting the importance of AB-cycle selection and offspring management. These findings offer practical guidance for crossover design, open avenues to improve Stage II strategies, and suggest broader implications for GPX and related recombination operators in large-scale TSP solving.

Abstract

The Edge Assembly Crossover (EAX) algorithm is the state-of-the-art heuristic for solving the Traveling Salesperson Problem (TSP). It regularly outperforms other methods, such as the Lin-Kernighan-Helsgaun heuristic (LKH), across diverse sets of TSP instances. Essentially, EAX employs a two-stage mechanism that focuses on improving the current solutions, first, at the local and, subsequently, at the global level. Although the second phase of the algorithm has been thoroughly studied, configured, and refined in the past, in particular, its first stage has hardly been examined. In this paper, we thus focus on the first stage of EAX and introduce a novel method that quickly verifies whether the AB-cycles, generated during its internal optimization procedure, yield valid tours -- or whether they need to be repaired. Knowledge of the latter is also particularly relevant before applying other powerful crossover operators such as the Generalized Partition Crossover (GPX). Based on our insights, we propose and evaluate several improved versions of EAX. According to our benchmark study across 10 000 different TSP instances, the most promising of our proposed EAX variants demonstrates improved computational efficiency and solution quality on previously rather difficult instances compared to the current state-of-the-art EAX algorithm.

Figures (9)

• Figure 1: Left: Parents $A$ (blue) and $B$ (orange) form $G_{AB}$, which contains all edges (bottom). Right: The AB-cycle (top), the (simplified) outer graphs (middle), and the invalid solution that would result from applying the AB-cycle to $A$.
• Figure 2: Left: Parent $A$ (blue, top) and Parent $B$ (orange, bottom) form $G_{AB}$, which contains all edges (middle). Right: Examples of AB-cycles resulting in different E-sets used by EAX. Points are distinguished into $A$- (blue), $B$- (red), and portals / $C$-vertices (black) in the corresponding E-set.
• Figure 3: Left: If the application of an AB-cycle leads to two or more subtours the solution is repaired by looking at the ten nearest neighbors of every node of the smallest subtour (here depicted by the dashed circle at the 4th nearest neighbor distance of the orange node) to find the most beneficial tuple of four nodes that remove two edges and add two edges to connect two subtours (right).
• Figure 4: Left: The total number of all children produced in stage I of EAX, differentiated by the AB-cycle type producing the offspring and distinguished by the instance sizes. On the $x$-axis are the number of portals or $C$-vertices in the AB-cycle and on the $y$-axis the number of subtours, which had to be joined to produce a valid solution. Right: The percentage of all offspring from the left side, which were selected to replace their respective parent $A$ (success rate). Interestingly, the AB-cycle type that is found most often with four portals and two subtours has one of the lowest success rates.
• Figure 5: Number of optimal edges gained vs. lost distinguished by the size (columns) and type (rows) of TSP set, as well as the number of portals (x-axis) and produced subtours (y-axis) per AB-cycle. Top: Ratio of all optimal edges introduced (vs. lost) by the specific AB-cycle type, ignoring edges that are removed or added in a potential repair mechanism. Bottom: Average ratio of all optimal edges introduced (vs. lost) during the corresponding repair mechanism. Notably, AB-cycle types that introduce relatively few subtours compared to the number of subtours that could potentially be generated appear to be favorable.