Approximation of minimum weight spanners for sparse graphs
نویسندگان
چکیده
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 graphG and an integer t , a t-spanner ofGwith theminimumnumber 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. © 2010 Elsevier B.V. All rights reserved.
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عنوان ژورنال:
- Theor. Comput. Sci.
دوره 412 شماره
صفحات -
تاریخ انتشار 2011