Pergamon 0960-0779(94)00284-3 Chaos, Solitons & Fractals Vol. 6, pp. 389-397, 1995 Elsevier Science Ltd Printed in Great Britain 0960-0779195 $9.50 + .00 The Euclidean Traveling Salesman Problem and a Space-Filling Curve MICHAEL G. NORMAN t and PABLO MOSCATO CeTAD, Universidad Nacional de La Plata, C.C. 75, 1900, La Plata, Argentina Abstract - We elucidate the relationship between space-fillling curves and the Euclidean Traveling: Salesman Problem (TSP) by reference to a particular space-filling curve whose scaling behaviour is strongly related to the conjectured scaling behaviour of the optimal TSP tour. We suggest that space-filling curves can be used to generate testbed TSPs: sets of points which in the limit cover a planar surface and for which tours of minimum length are known. INTRODUCTION In this paper we discuss a way to generate large problem instances of TSP, with unique global optima. This approach is complementary to TSPLIB--a public domain database, accessible via ftp 3, which contains many large TSP instances solved to optimality through the use of exact methods like branch-and-cut. We suggest that space-filling curves can be used to generate large TSP instances which, as well as being cleanly defined, can provide significant insight on the characteristics of the TSP and the nature of heuristic approximation methods. It has been noted elsewhere [1] that there is a strong relationship between the well defined notion of computational complexity, see for example [2], and the less well defined notion of complexity which is used in the literature describing fractals, chaos and emergent behaviour. Whilst, this paper does not make a formal connection between the two notions, we take the opportunity of pointing out specific connections which apertain to our particular problem. NPEANO In the widely-disseminated freeware software package called FRACTINT (Version 17.2) there is a program developed by A. Mariano which can be used to generate fractals by interpretation of L-systems [3, 1Current address, and address for correspondence: MakespanLimited, 8 GayfieldSquare, Edinburgh EH1 3NT, Scotland 2US spelling is used by convention. 3TSPLIB is compiled and maintained by Gerhard Reinelt of the "Institut fuer Mathematik, Universitaet Augsburg", E-Mail reinel~@augsopt .uni-augsburg. de. 389