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Directed Steiner Tree and the Lasserre Hierarchy (1111.5473v2)
Published 23 Nov 2011 in cs.DS
Abstract: The goal for the Directed Steiner Tree problem is to find a minimum cost tree in a directed graph G=(V,E) that connects all terminals X to a given root r. It is well known that modulo a logarithmic factor it suffices to consider acyclic graphs where the nodes are arranged in L <= log |X| levels. Unfortunately the natural LP formulation has a |X|1/2 integrality gap already for 5 levels. We show that for every L, the O(L)-round Lasserre Strengthening of this LP has integrality gap O(L log |X|). This provides a polynomial time |X|{epsilon}-approximation and a O(log3 |X|) approximation in O(n{log |X|) time, matching the best known approximation guarantee obtained by a greedy algorithm of Charikar et al.
- Thomas Rothvoß (7 papers)