Construction of efficient communication sub-structures: Non-approximability results and polynomial sub-cases

Abstract

In this paper, we study the following problem. Given a graph G = (V,E) and M ⊆ V, construct a subgraph G*"M = (V *,E*) of G spanning M, with the minimum number of edges and such that for all u, v ∈ M the distance between u and v in G and in G*M is the same. This is what we call an optimal partial spanner of M in G. Such a structure is "between" a Steiner tree and a spanner and could be a particulary performant and low cost structure connecting members in a network. We prove that the problem cannot be approximated within a constant factor. We then focus on special cases: We require that the partial spanner is a tree satisfying additionnai conditions. For this sub problem, we describe a polynomial algorithm to construct such a tree partial spanner. © Springer-Verlag 2003.