Rethinking Neural Combinatorial Optimization for Vehicle Routing Problems with Different Constraint Tightness Degrees

TL;DR

This work reveals that neural combinatorial optimization methods for vehicle routing overfit to a fixed constraint tightness, such as CVRP capacity $C$, and perform poorly on out-of-domain tightness levels. It introduces a straightforward training scheme that exposes models to a range of tightness values (VCT) and a multi-expert module (MEM) that specializes strategies by tightness range. Empirical results on CVRP and CVRPTW show significant robustness and performance gains, with average optimality gaps dropping from around $7{-}10 ext{\%}$ to below $2 ext{\%}$ across diverse constraint degrees. The approach enhances practical applicability of NCO to real-world VRPs where constraint tightness varies, and suggests avenues for adaptive, continuously generalized policies in COPs.

Abstract

Recent neural combinatorial optimization (NCO) methods have shown promising problem-solving ability without requiring domain-specific expertise. Most existing NCO methods use training and testing data with a fixed constraint value and lack research on the effect of constraint tightness on the performance of NCO methods. This paper takes the capacity-constrained vehicle routing problem (CVRP) as an example to empirically analyze the NCO performance under different tightness degrees of the capacity constraint. Our analysis reveals that existing NCO methods overfit the capacity constraint, and they can only perform satisfactorily on a small range of the constraint values but poorly on other values. To tackle this drawback of existing NCO methods, we develop an efficient training scheme that explicitly considers varying degrees of constraint tightness and proposes a multi-expert module to learn a generally adaptable solving strategy. Experimental results show that the proposed method can effectively overcome the overfitting issue, demonstrating superior performances on the CVRP and CVRP with time windows (CVRPTW) with various constraint tightness degrees.

Figures (3)

• Figure 1: Impact of Capacity on Problem Similarity. The CVRP100 solution closely resembles the OVRP solution when capacity is very small (e.g., C=10) and becomes equivalent to the TSP solution when capacity is extremely large (e.g., C=500). We systematically quantify the problem similarity in Appendix \ref{['Problem Similarity']}.
• Figure 2: Performance of different NCO methods on CVRP100 instances with extreme constraint tightness degrees (i.e., C=10/500). "-OVRP", "-TSP", and "-CVRP" indicate that the methods are specifically designed to solve OVRP, TSP, and CVRP instances, respectively. A smaller optimality gap indicates better performance.
• Figure 3: Model structure featuring a multi-expert module. The model adopts a light encoder and heavy decoder architecture, incorporating an effective multi-expert module within the decoder. The multi-expert module consists of a gate mechanism and $m$ expert layers. The gate mechanism selects instances within specific ranges of constraint tightness to their corresponding expert layer for processing. Each expert layer processes the node embeddings from the $L$ stacked attention layers and generates probability distributions for node selection during solution construction.