A t-spanner of a graph G is its spanning subgraph S such that the distance between every pair of vertices in S is at most t times their distance in G. The sparsest t-spanner problem asks to find, for a given graph G and an integer t, a t-spanner of G with the minimum number of edges. 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 the approximability status of the problem on planar graphs was open. We resolve this open issue by showing that the sparsest t-spanner problem admits the efficient polynomial time approximation scheme (EPTAS) for every t ≥ 1. Our result holds for a much wider class of graphs, namely, the class of apex-minor-free graphs, which contains the classes of planar and bounded genus graphs. Moreover, it is possible to extend our results to weighted apex-minor free graphs, when the maximum edge weight is bounded by some constant.
Dragan et al. REF gave an EPTAS for the problem in planar graphs.
Approximation of minimum weight spanners for sparse graphs
Theor. Comput. Sci.
Theor. Comput. Sci.
Computer Science Mathematics