5 research outputs found
Directed Steiner Tree and the Lasserre Hierarchy
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(log^3 |X|) approximation in O(n^{log |X|)
time, matching the best known approximation guarantee obtained by a greedy
algorithm of Charikar et al.Comment: 23 pages, 1 figur
Integrality gaps of semidefinite programs for Vertex Cover and relations to embeddability of Negative Type metrics
We study various SDP formulations for {\sc Vertex Cover} by adding different
constraints to the standard formulation. We show that {\sc Vertex Cover} cannot
be approximated better than even when we add the so called pentagonal
inequality constraints to the standard SDP formulation, en route answering an
open question of Karakostas~\cite{Karakostas}. We further show the surprising
fact that by strengthening the SDP with the (intractable) requirement that the
metric interpretation of the solution is an metric, we get an exact
relaxation (integrality gap is 1), and on the other hand if the solution is
arbitrarily close to being embeddable, the integrality gap may be as
big as . Finally, inspired by the above findings, we use ideas from the
integrality gap construction of Charikar \cite{Char02} to provide a family of
simple examples for negative type metrics that cannot be embedded into
with distortion better than 8/7-\eps. To this end we prove a new
isoperimetric inequality for the hypercube.Comment: A more complete version. Changed order of results. A complete proof
of (current) Theorem
Interactive proof system variants and approximation algorithms for optical networks
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1996.Includes bibliographical references (p. 111-121).by Ravi Sundaram.Ph.D
Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)
The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..