# Directed Steiner Tree and the Lasserre Hierarchy

@article{Rothvoss2011DirectedST, title={Directed Steiner Tree and the Lasserre Hierarchy}, author={Thomas Rothvoss}, journal={ArXiv}, year={2011}, volume={abs/1111.5473} }

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…

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## 38 Citations

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## References

SHOWING 1-10 OF 49 REFERENCES

Integrality ratio for group Steiner trees and directed steiner trees

- Computer Science, MathematicsSODA '03
- 2003

An Ω(log2k) lower bound on the integrality ratio of the flow-based relaxation for the Group Steiner Tree problem is presented, where k denotes the number of groups; this holds even for input graphs that are Hierarchically Well-Separated Trees, in which case this lower bound is tight.

Approximation algorithms for directed Steiner problems

- Mathematics, Computer ScienceSODA '98
- 1998

We obtain the first non-trivial approximation algorithms for the Steiner Tree problem and the Generalized Steiner Tree problem in general directed graphs. Essentially no approximation algorithms were…

The Polymatroid Steiner Problems

- MathematicsJ. Comb. Optim.
- 2005

The Polymatroid Steiner Problem, in which a polymatroid P = P(V) is defined on V and the Steiner tree is required to span at least one base of P, is considered, which can be approximately solved by algorithms generalizing methods of Chekuri et al. (2002).

A series of approximation algorithms for the acyclic directed steiner tree problem

- Computer ScienceAlgorithmica
- 2006

This paper gives anO(kε)-approximation algorithm for any ε>0.1, which improves the previously knownk-approximating.

A polylogarithmic approximation algorithm for the group Steiner tree problem

- Mathematics, Computer ScienceSODA '98
- 1998

A randomized algorithm with a polylogarithmic approximation guarantee for the group Steiner tree problem is given and improves existing approximation results for network design problems with location-based constraints and for the symmetric generalized traveling salesman problem.

Nearly complete graphs decomposable into large induced matchings and their applications

- MathematicsSTOC '12
- 2012

Two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings are described, which disproves (in a strong form) a conjecture of Meshulam, substantially improves a result of Birk, Linial andMeshulam on communicating over a shared channel, and extends the analysis of Hastad and Wigderson of the graph test of Samorodnitsky and Trevisan for linearity.

Polylogarithmic inapproximability

- Computer Science, MathematicsSTOC '03
- 2003

It is shown that for every fixed ε>0, the GROUP-STEINER-TREE problem admits no efficient log2-ε k approximation, where k denotes the number of groups (or, alternatively, the input size), unless NP has quasi polynomial Las-Vegas algorithms.

New Tools for Graph Coloring

- Computer ScienceAPPROX-RANDOM
- 2011

This algorithm is inspired by recent work of Barak, Raghavendra, and Steurer on using Lasserre Hierarchy for unique games and can be used to show that known integrality gap instances for SDP relaxations like strict vector chromatic number cannot survive a few rounds of Lasserr lifting.

Linear Level Lasserre Lower Bounds for Certain k-CSPs

- Mathematics, Computer Science2008 49th Annual IEEE Symposium on Foundations of Computer Science
- 2008

We show that for kges3 even the Omega(n) level of the Lasserre hierarchy cannot disprove a random k-CSP instance over any predicate type implied by k-XOR constraints, for example k-SAT or k-XOR. (One…

Lasserre Hierarchy, Higher Eigenvalues, and Approximation Schemes for Graph Partitioning and Quadratic Integer Programming with PSD Objectives

- Computer Science, Mathematics2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
- 2011

An approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semi definite objective functions and global linear constraints is presented, and an algorithm for independent sets in graphs that performs well when the Laplacian does not have too many eigenvalues bigger than $1+o(1).