A Consensus-Edge Initializer for Multi-start 2-opt on the Symmetric Euclidean Travelling Salesman Problem

We study a one-shot consensus-based initializer for multi-start 2-opt on symmetric Euclidean Travelling Salesman Problem (TSP) instances. Given the M local minima from best-of-M 2-opt, we count undirected edge frequencies, greedily assemble a tour from the most frequent edges with a length-biased tie-break, and run a single additional 2-opt descent from this seed. The local search operator is unchanged; the consensus stage reuses existing tours and adds only a small overhead beyond the M descents, and we characterise this overhead analytically while focusing evaluation on tour length improvements. On synthetic Euclidean instances from three generators, we evaluate performance over a grid of problem sizes n and restart budgets M. The consensus + 2-opt pipeline improves best-of-M 2-opt by roughly 3–6% in median tour length with high win rates for n≥500 and moderate M, while for small n and large M the baseline saturates and gains shrink to zero or slightly negative values. Thus, even a single aggregation step over 2-opt local minima yields a simple, reproducible seed that is beneficial in the practically relevant regime of moderate budgets and larger instances and approximately neutral elsewhere.