The Time-Dependent Traveling Salesman Problem (TDTSP) extends the classical TSP by allowing dynamic edge weights that vary with departure time, reflecting real-world scenarios such as transportation networks, where travel times fluctuate due to congestion patterns. TDTSP violates symmetry, triangle inequality, and cyclic invariance properties of classical TSP, creating unique computational challenges. In this paper, we propose a neural model that extends MatNet from static asymmetric TSP to time-dependent settings by using an adjacency tensor to capture temporal variations, followed by a time-aware decoder. Our architecture addresses the unique challenge of asymmetry and triangle inequality violations that change dynamically over time. Beyond architectural innovations, our research reveals a critical evaluation insight: many practical TDTSP instances maintain the same optimal solution regardless of time-dependent edge weights.

The Asymmetric Traveling Salesman Problem (ATSP) is one of the most fundamental and notoriously challenging problems in combinatorial optimization. We propose a novel continuous relaxation framework for ATSP that leverages differentiable constraints to encourage acyclic structures and valid permutations.

We propose a novel continuous relaxation framework for the Asymmetric Traveling Salesman Problem (ATSP) by leveraging differentiable constraints that encourage acyclic structures and valid permutations.

We consider the problem of inferring high-dimensional datax in a model that consists of a priorp(x) and an auxiliary differentiable constraintc(x,y) on x given some additional informationy. In this paper, the prior is an independently trained denoising diffusion generative model. The auxiliary constraint is expected to have a differentiable form, but can come from diverse sources.

The seeder generates as diversified candidate solutions as possible (seeds) while being dedicated to exploring over the full combinatorial action space (i.e.,sequence ofassignment action).

Recently, deep reinforcement learning (DRL) models have shown promising results in solving NP-hard Combinatorial Optimization (CO) problems. However, most DRL solvers can only scale to a few hundreds of nodes for combinatorial optimization problems on graphs, such as the Traveling Salesman Problem (TSP). This paper addresses the scalability challenge in large-scale combinatorial optimization by proposing a novel approach, namely, DIMES. Unlike previous DRL methods which suffer from costly autoregressive decoding or iterative refinements of discrete solutions, DIMES introduces a compact continuous space for parameterizing the underlying distribution of candidate solutions. Such a continuous space allows stable REINFORCE-based training and fine-tuning via massively parallel sampling. We further propose a meta-learning framework to enable the effective initialization of model parameters in the fine-tuning stage. Extensive experiments show that DIMES outperforms recent DRL-based methods on large benchmark datasets for Traveling Salesman Problems and Maximal Independent Set problems.

We present NeuroLKH, a novel algorithm that combines deep learning with the strong traditional heuristic Lin-Kernighan-Helsgaun (LKH) for solving Traveling Salesman Problem. Specifically, we train a Sparse Graph Network (SGN) with supervised learning for edge scores and unsupervised learning for node penalties, both of which are critical for improving the performance of LKH. Based on the output of SGN, NeuroLKH creates the edge candidate set and transforms edge distances to guide the searching process of LKH. Extensive experiments firmly demonstrate that, by training one model on a wide range of problem sizes, NeuroLKH significantly outperforms LKH and generalizes well to much larger sizes. Also, we show that NeuroLKH can be applied to other routing problems such as Capacitated Vehicle Routing Problem (CVRP), Pickup and Delivery Problem (PDP), and CVRP with Time Windows (CVRPTW).