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Look-Ahead and Degree-Aware Branch-and-Bound for the Symmetric Traveling Salesman Problem: Algorithm, Complexity, and the Decisive Role of the Lower Bound

LDA‑B&B is a branch‑and‑bound solver for symmetric TSP combining multi‑start nearest‑neighbour/2‑opt incumbents, look‑ahead and regret branch ordering, and a residual MST lower bound. The implementation contains a critical bug: the MST routine silently prunes the root node, returning a heuristic tour as proven optimal. After fixing this, the solver is exact but only up to ~20 cities. Look‑ahead and regret change node counts by 0.4% when optimality is proven. Replacing the MST bound with a Held–Karp 1‑tree bound dramatically improves performance: it reduces root gaps from 20–36% to 1% and proves random instances with 25 and 30 cities in 301 and 536 nodes. All eight small TSPLIB instances (n ≤ 42) are solved to optimality in under 4 seconds. Root bounds on mid‑size instances (1 060–2 319 cities) lie 0.5–2.2% below reference optima. Clustered instances remain hard. The study concludes that the lower bound, not branching heuristics, determines the exactness frontier.

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