The Generalized Travelling Salesman Problem, denoted by GTSP, is a variant of the classical travelling salesman problem (TSP), in which the nodes of an undirected graph are partitioned into node sets (clusters) and the salesman has to visit exactly one node from every cluster. In this paper we described six distinct formulations of the GTSP as an integer programming. Apart from the standard for...

We provide a uniform estimate for the L-norm (over any interval of bounded length) of the logarithmic derivatives of global normalizing factors associated to intertwining operators for the following reductive groups over number fields: inner forms of GL(n); quasi-split classical groups and their similitude groups; the exceptional group G2. This estimate is a key ingredient in the analysis of th...

Let tsp(G) denote the length of a shortest travelling salesman tour in a graph G. We prove that for any ε > 0, there exists a simple 2-connected planar cubic graph G1 such that tsp(G1) ≥ (1.25 − ε) · |V (G1)|, a simple 2-connected bipartite cubic graph G2 such that tsp(G2) ≥ (1.2 − ε) · |V (G2)|, and a simple 3-connected cubic graph G3 such that tsp(G3) ≥ (1.125− ε) · |V (G3)|.

In this paper we present a new Ants System approach to a dynamic Travelling Salesman Problem. Here the travel times between the cities are subject to change. To handle this dynamism several ways of adapting the pheromone matrix both locally and globally are considered. We show that the strategy of smoothing pheromone values only in the area containing a change leads to improved results.

Human visual/spatial problem solving often requires both global and local information to be processed. But the relationship between those two kinds of information and the way in which they interact with one another during problem solving has not been thoroughly discussed. In the particular setting of solving the traveling salesman problem (TSP), we investigated into the relative roles of global...

A new method due to Fokas for explicitly solving boundary-value problems for linear partial differential equations is extended to equations with mixed partial derivatives. The Benjamin-Bona-Mahony equation is used as an example: we consider the Robin problem for this equation posed both on the half line and on the finite interval. For specific cases of the Robin boundary conditions the boundary...

We study the problem of exploring all nodes of an unknown directed graph. A searcher has to construct a tour that visits all nodes, but only has information about the parts of the graph it already visited. Analogously to the travelling salesman problem, the goal is to minimize the cost of such a tour. In this article, we present upper and lower bounds for the competitive ratio of both the deter...

We consider online versions of the Traveling Salesman Problem (TSP) and Traveling Repairman Problem (TRP) where instances are not known in advance. Cities (points) to be visited are revealed over time, while the server is en route serving previously released requests. These problems are known in the literature as the Online TSP (TRP, respectively). The corresponding offline problems are the TSP...

The n-card problem is to determine the minimal intervals [u, v] such that for every n×n stochastic matrix A there is an n×n permutation matrix P (depending on A) such that tr(PA) ∈ [u, v]. This problem is closely related to classical mathematical problems from industry and management, including the linear assignment problem and the travelling salesman problem. The minimal intervals for the n-ca...