A First Guess is Rarely the Final Answer: Learning to Search in the Travelling Salesperson Problem

Most neural solvers for the Traveling Salesperson Problem (TSP) are trained to output a single solution, even though practitioners rarely stop there: at test time, they routinely spend extra compute on sampling or post-hoc search. This raises a natural question: can the search procedure itself be learned? Neural improvement methods take this perspective by learning a policy that applies local modifications to a candidate solution, accumulating gains over an improvement trajectory. Yet learned improvement for TSP remains comparatively immature, with existing methods still falling short of robust, scalable performance. We argue that a key reason is design mismatch: many approaches reuse state representations, architectural choices, and training recipes inherited from single-solution methods, rather than being built around the mechanics of local search. This mismatch motivates NICO-TSP (Neural Improvement for Combinatorial Optimization): a 2-opt improvement framework for TSP. NICO-TSP represents the current tour with exactly $n$ edge tokens aligned with the neighborhood operator, scores 2-opt moves directly without tour positional encodings, and trains via a two-stage procedure: imitation learning to short-horizon optimal trajectories, followed by critic-free group-based reinforcement learning over longer rollouts. Under compute-matched evaluations that measure improvement as a function of both search steps and wall-clock time, NICO-TSP delivers consistently stronger and markedly more step-efficient improvement than prior learned and heuristic search baselines, generalizes far more reliably to larger out-of-distribution instances, and serves both as a competitive replacement for classical local search and as a powerful test-time refinement module for constructive solvers.

Enabling Population-Based Architectures for Neural Combinatorial Optimization

Neural Combinatorial Optimization (NCO) has mostly focused on learning policies, typically neural networks, that operate on a single candidate solution at a time, either by constructing one from scratch or iteratively improving it. In contrast, decades of work in metaheuristics have shown that maintaining and evolving populations of solutions improves robustness and exploration, and often leads to stronger performance. To close this gap, we study how to make NCO explicitly population-based by learning policies that act on sets of candidate solutions. We first propose a simple taxonomy of population awareness levels and use it to highlight two key design challenges: (i) how to represent a whole population inside a neural network, and (ii) how to learn population dynamics that balance intensification (generating good solutions) and diversification (maintaining variety). We make these ideas concrete with two complementary tools: one that improves existing solutions using information shared across the whole population, and the other generates new candidate solutions that explicitly balance being high-quality with diversity. Experimental results on Maximum Cut and Maximum Independent Set indicate that incorporating population structure is advantageous for learned optimization methods and opens new connections between NCO and classical population-based search.