Directed Steiner Tree and the Lasserre Hierarchy

TL;DR

This paper demonstrates that applying the Lasserre hierarchy to the Directed Steiner Tree problem yields significantly improved approximation guarantees, bridging the gap between LP relaxations and optimal solutions.

Contribution

It shows that the Lasserre hierarchy can be used to obtain better approximation ratios for the Directed Steiner Tree problem than previous LP-based methods.

Findings

Lasserre hierarchy reduces integrality gap to O(L log |X|) for any L.

Provides polynomial-time approximation of |X|^{epsilon}.

Achieves O(log^3 |X|) approximation matching previous greedy algorithms.

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(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.