The multi-stripe travelling salesman problem

In the classical Travelling Salesman Problem (TSP), the objective function sums the costs for travelling from one city to the next city along the tour. In the q-stripe TSP with q ≥ 1, the objective function sums the costs for travelling from one city to each of the next q cities along the tour. The resulting q-stripe TSP generalizes the TSP and forms a special case of the quadratic assignment problem. We analyze the computational complexity of the q-stripe TSP for various classes of specially structured distance matrices. We derive NP-hardness results as well as polyomially solvable cases. One of our main results generalizes a well-known theorem of Kalmanson from the classical TSP to the q-stripe TSP

A note on the bottleneck graph partition problem

The bottleneck graph partition problem consists of partitioning the vertices of an undirected edge-weighted graph into two equally sized sets such that the maximum edge weight in the cut separating the two sets becomes minimum. In this short note, we present an optimum algorithm for this problem with running time O(n2), where n is the number of vertices in the graph. Our result answers an open problem posed in a recent paper by Hochbaum and Pathria (1996)

The Steiner tree problem in Kalmanson matrices and in circulant matrices

We investigate the computational complexity of two special cases of the Steiner tree problem where the distance matrix is a Kalmanson matrix or a circulant matrix, respectively. For Kalmanson matrices we develop an efficient polynomial time algorithm that is based on dynamic programming. For circulant matrices we give an NP -hardness proof and thus establish computational intractability

A note on the bottleneck graph partition problem

The bottleneck graph partition problem consists of partitioning the vertices of an undirected edge-weighted graph into two equally sized sets such that the maximum edge weight in the cut separating the two sets becomes minimum. In this short note, we present an optimum algorithm for this problem with running time O(n2), where n is the number of vertices in the graph. Our result answers an open problem posed in a recent paper by Hochbaum and Pathria (1996)

Faster algorithms for computing power indices in weighted voting games

We consider weighted voting games with n players. We show how to compute the Banzhaf power index for every player within a running time of O(n2 1.415n), and how to compute the Shapley–Shubik power index within a running time of O(n 1.415n). Our result improves on the straightforward running times of O(n2 2n) and O(n 2n), respectively, that are implicit in the definitions of these power indices