A simple Path-based LP Relaxation for Directed Steiner Tree

TL;DR

This paper introduces a straightforward path-based LP relaxation for the Directed Steiner Tree problem in layered graphs, achieving a near-optimal integrality gap and simplifying previous complex hierarchical approaches.

Contribution

It presents a simple LP relaxation that matches the best known bounds for DST, bypassing hierarchy machinery and providing a clearer proof of integrality gap bounds.

Findings

Achieves an O(l log k) integrality gap for DST

Simplifies previous hierarchy-based proofs

Provides an alternative proof using Sherali-Adams hierarchy

Abstract

We study the Directed Steiner Tree (DST) problem in layered graphs through a simple path-based linear programming relaxation. This relaxation achieves an integrality gap of O(l log k), where k is the number of terminals and l is the number of layers, which matches the best known bounds for DST previously obtained via lift-and-project hierarchies. Our formulation bypasses hierarchy machinery, offering a more transparent route to the state-of-the-art bound, and it can be exploited to provide an alternative simpler proof that O(l) rounds of the Sherali-Adams hierarchy suffice for reducing the integrality gap on layered instances of DST.