Summary: We consider generalizations of the \(k\)-source sum of vertex eccentricity problem (\(k\)-SVET) and the \(k\)-source sum of source eccentricity problem (\(k\)-SSET) [H. S. Connamacher and A. Proskurowski, “The complexity of minimizing certain cost metrics for \(k\)-source spanning trees”, Discrete Appl. Math. 131, No. 1, 113–127 (2003; Zbl 1073.68061)], which we call SDET and SSET, respectively, and provide efficient algorithms for their solution. The SDET (SSET, resp.) problem is defined as follows: given a weighted graph \(G\) and sets \(S\) of source nodes and \(D\) of destination nodes, which are subsets of the vertex set of \(G\), construct a tree-subgraph \(T\) of \(G\) which connects all sources and destinations and minimizes the SDET cost function \(\sum_{d \in D}\max_{s \in S}d_{T}(s,d)\) (the SSET cost function \(\sum_{s \in S}\max_{d\in D}d_T(s,d)\), respectively). We describe an \(O(nm\log n)\)-time algorithm for the SDET problem and thus, by symmetry, to the SSET problem, where \(n\) and \(m\) are the numbers of vertices and edges in \(G\). The algorithm introduces efficient ways to identify candidates for the sought tree and to narrow down their number to \(O(m)\). Our algorithm readily implies \(O(nm\log n)\)-time algorithms for the \(k\)-SVET and \(k\)-SSET problems as well.
For the entire collection see [Zbl 1098.68001].