Estimating the Held–Karp lower bound for the geometric travelling salesman problem

The Held–Karp lower bound (HK) provides a very good problem-specific estimate of optimal tour length for the travelling salesman problem (TSP). This measure, which is relatively quick and easy to compute, has enormous practical value when evaluating the quality of near optimal solutions for large problems where the true optima are not known. Although HK can be evaluated exactly by Linear Programming techniques, code for doing this efficiently for problems larger than a few hundred cities is not readily available or easy to produce. In addition Linear Programming implementations (even efficient ones) do not scale well and rapidly become impractical for problems with many thousands of cities. In this study we concentrate on the iterative estimation approach proposed by Held and Karp in their original papers. Our paper provides implementation details for two iterative schemes which both use the subgraph speed-up technique. We begin by surveying the important theoretical issues which underpin the original iterative approach of Held and Karp (now known as subgradient optimisation). We then present some detailed practical guidelines for the evaluation of HK for large geometric TSPs, and also provide some empirical evidence demonstrating the robustness of the iterative schemes we have used. Finally our estimation of the Goemans and Bertsimas constant provides an independent confirmation of the value published recently by Johnson, McGeoch and Rothberg and simultaneously supports the claim that our approach does indeed produce reliable results.