Given a directed graph G with non-negative cost on the arcs, a directed tour cover T ofG is a cycle (not necessarily simple) inG such that either head or tail (or both of them) of every arc in G is touched by T . The minimum directed tour cover problem (DToCP), which is to find a directed tour cover of minimum cost, is NP -hard. It is thus interesting to design approximation algorithms with per...

The Covering Steiner problem is a common generalization of the k-MST and Group Steiner problems. An instance of the Covering Steiner problem consists of an undirected graph with edge-costs, and some subsets of vertices called groups, with each group being equipped with a non-negative integer value (called its requirement); the problem is to find a minimum-cost tree which spans at least the requ...

In 1914 Lebesgue defined a ‘universal covering’ to be a convex subset of the plane that contains an isometric copy of any subset of diameter 1. His challenge of finding a universal covering with the least possible area has been addressed by various mathematicians: Pál, Sprague and Hansen have each created a smaller universal covering by removing regions from those known before. However, Hansen’...

The traveling salesman problem is a classical NP-hard problem, and also hard to approximate at all under the assumption P 6= NP. Definition 1. A vertex tour of a graph G is a path which visits all vertices and returns to its starting vertex. This is another name for a Hamiltonian cycle on G. Traveling salesman problems involve finding a tour of undirected graph G with edge weights d(·), which w...

1994 Zentrum f ur Paralleles Rechnen Universit at zu K oln Weyertal 80 D{50931 K oln Abstract We describe an improved clustering heuristic for the Eucledian Traveling Salesman Problem and its parallelization for a distributed memory machine. A geometric decomposition is used for the clustering{stage and special emphasis has been put on the computation of the global tour through the clusters. Th...