An Approximation Algorithm for the Tree t-Spanner Problem on Unweighted Graphs via Generalized Chordal Graphs

Abstract

A spanning tree T of a graph G is called a tree t-spanner of G if the distance between every pair of vertices in T is at most t times their distance in G. In this paper, we present an algorithm which constructs for an n-vertex m-edge unweighted graph G: (1) a tree $(2\lfloor\log_2 n\rfloor)$-spanner in O(mlogn) time, if G is a chordal graph; (2) a tree $(2\rho\lfloor\log_2 n\rfloor)$-spanner in O(mnlog² n) time or a tree $(12\rho\lfloor\log_2 n\rfloor)$-spanner in O(mlogn) time, if G is a graph admitting a Robertson-Seymour’s tree-decomposition with bags of radius at most ρ in G; and (3) a tree $(2\lceil{t/2}\rceil\lfloor\log_2 n\rfloor)$-spanner in O(mnlog² n) time or a tree $(6t\lfloor\log_2 n\rfloor)$-spanner in O(mlogn) time, if G is an arbitrary graph admitting a tree t-spanner. For the latter result we use a new necessary condition for a graph to have a tree t-spanner: if a graph G has a tree t-spanner, then G admits a Robertson-Seymour’s tree-decomposition with bags of radius at most ⌈t/2⌉ in G.