Learning to Handle Complex Constraints for Vehicle Routing Problems

TL;DR

This paper proposes a novel Proactive Infeasibility Prevention (PIP) framework, which integrates the Lagrangian multiplier as a basis to enhance constraint awareness and introduces preventative infeasibility masking to proactively steer the solution construction process.

Abstract

Vehicle Routing Problems (VRPs) can model many real-world scenarios and often involve complex constraints. While recent neural methods excel in constructing solutions based on feasibility masking, they struggle with handling complex constraints, especially when obtaining the masking itself is NP-hard. In this paper, we propose a novel Proactive Infeasibility Prevention (PIP) framework to advance the capabilities of neural methods towards more complex VRPs. Our PIP integrates the Lagrangian multiplier as a basis to enhance constraint awareness and introduces preventative infeasibility masking to proactively steer the solution construction process. Moreover, we present PIP-D, which employs an auxiliary decoder and two adaptive strategies to learn and predict these tailored masks, potentially enhancing performance while significantly reducing computational costs during training. To verify our PIP designs, we conduct extensive experiments on the highly challenging Traveling Salesman Problem with Time Window (TSPTW), and TSP with Draft Limit (TSPDL) variants under different constraint hardness levels. Notably, our PIP is generic to boost many neural methods, and exhibits both a significant reduction in infeasible rate and a substantial improvement in solution quality.

Figures (11)

• Figure 1: A TSPTW instance to illustrate the malfunction of existing masking mechanism (left three panels) and NP-hardness of obtaining precise infeasible masks (right panel). The orange bar represents the time window [$l_i$, $u_i$] for node $v_i$. For the partial solution $v_0 \rightarrow v_1$, both $v_2$ and $v_3$ are locally feasible. However, selecting $v_3$ results in the irreversible infeasibility of $v_2$ afterwards.
• Figure 2: Illustration of policy optimization trajectories on VRP with varying difficulty levels - (a)(b)(d) easy and (c)(e) hard, and different constraint handling schemes - (a) feasibility masking, (b)(c) Lagrangian multiplier, and (d)(e) our PIP. The orange-filled circle denotes the feasible policy space $\Pi_F$, while the dotted frame represents the actual search space of the neural policies $\pi_{\theta}$.
• Figure 3: An illustrative overview of our proposed approach: Left - Preventative infeasibility (PI) estimator. Right - PIP (highlighted in green) framework and PIP-D (highlighted in blue) framework.
• Figure 4: Effects of $\mathcal{J}_{\text{IN}}$
• Figure 5: Effects of Less Update.