Mathematical Methods of Operations Research (1996) 43 : 169-181 Maximum Travelling Salesman Problem VLADIMIR KOTOV AND MICHAIL KOVALEV1 Belarussian State University, Faculty of Applied Mathematics and Informatics, Prospekt F. Skarina 4, 220050 Minsk, Belarus e-maih on.kovalev@zib-berlin.de Abstract: We present different types of techniques for designing algorithms with worst-case perlbr- mances for the Maximum Travelling Salesman Problem. Key Words: Travelling Salesman Problem, approximation algorithms, worst-case performance. 1 Introduction The maximum travelling salesman problem (MTSP) is formulated as follows: In a complete graph K, with non-negative edge weights we look for a Hamil- tonian circuit (directed circuit) with maximal total weight. For undirected graphs we call this problem the symmetric (or shortly SMTSP) and for directed graphs asymmetric (shortly AMTSP), respectively. If the triangle inequality holds for all nodes, we call the problem metric MTSP (MMTSP). The weight of edge e = (i,j) is denoted by c(e) or cq respectively. For a collection X of edges, we denote by c(X) the sum of the weight of edges in X. Our objective is to compute worst-case performances of algorithms that are valid for all problem instances and have a form c(X ~') >_ c~ 9c(X ~ where H refers to one of the heuristics (greedy, matching) or relaxations (2- matching, assignment) or their combinations. At first we present the greedy algorithm with a subtour as starting point. We define a subtour to be a set of edges which is is contained in a Hamiltonian 1 Supported by Byelarussian Fundamental Science Found and DAAD 0340- 9422/96/43 : 2/169-181 $2.50 9 1996 Physica-Verlag, Heidelberg