A PTAS for the Sparsest Spanners Problem on Apex-Minor-Free Graphs

A t-spanner of a graph G is a spanning subgraph S in which the distance between every pair of vertices is at most t times their distance in G. The sparsest t-spanner problem asks to find, for a given graph G andanintegert, at-spanner of G with the minimum number of edges. On general n-vertex graphs, the problem is known to be NP-hard for all t ≥ 2, and, even more, it is NP-hard to approximate it with ratio O(log n) for every t ≥ 2. For t ≥ 5, the problem remains NP-hard for planar graphs, and up to now the approximability status of the problem on planar graphs considered to be open. In this note, we resolve this open issue by showing that the sparsest t-spanner problem admits a polynomial time approximation scheme (PTAS) for every t ≥ 1. Actually, our results hold for a much wider class of graphs, namely, on the class of apex-minor-free graphs which contains the classes of planar and bounded genus graphs