In the rectilinear Steiner arborescence problem the task is to build a shortest rectilinear Steiner tree connecting a given root and a set of terminals which are placed in the plane such that all root-terminal-paths are shortest paths. This problem is known to be NP-hard. In this paper we consider a more restricted version of this problem. In our case we have a depth restrictions d(t) ∈ N for e...

Let G be a connected graph. A vertex w strongly resolves a pair u, v of vertices of G if there exists some shortest u − w path containing v or some shortest v − w path containing u. A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W . The smallest cardinality of a strong resolving set for G is called the strong metric dime...

A path in a vertex-colored graph G is vertex rainbow if all of its internal vertices have a distinct color. The graph G is said to be rainbow vertex connected if there is a vertex rainbow path between every pair of its vertices. Similarly, the graph G is strongly rainbow vertex connected if there is a shortest path which is vertex rainbow between every pair of its vertices. We consider the comp...

The shortest path problem is considered with yet another extension. The transit times b(x,y,u) and the arc costs c(x,y,u) on arc (x,y) vary as a function of the departure time u at the beginning vertex of the arc. The problem considered is to find the shortest path between a pair of nodes such that the total traverse time is lesser than some specified value T. Waiting times at nodes are conside...

We investigate the uncertain versions of two classical combinatorial optimization problems, namely the Single-Pair Shortest Path Problem (SP-SPP) and the Single-Source Shortest Path Problem (SS-SPP). The former consists of finding a path of minimum length connecting two specific nodes in a finite directed graph G; the latter consists of finding the shortest paths from a fixed node to the remain...

We consider two orientation problems in a graph, namely the minimization of the sum of all the shortest path lengths and the minimization of the diameter. We show that it is NP-complete to decide whether a graph has an orientation such that the sum of all the shortest paths lengths is at most an integer specified in the input. The proof is a short reduction from a result of Chvátal and Thomasse...

Under the Open Shortest Path First (OSPF) protocol, traffic flow in an Internet Protocol (IP) network is routed on the shortest paths between each source and destination. The shortest path is calculated based on pre-assigned weights on the network links. The OSPF weight setting problem is to determine a set of weights such that, if the flow is routed based on the OSPF protocol, some measure of ...