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A simple Path-based LP Relaxation for Directed Steiner Tree

Kanstantsin Pashkovich, Marta Pozzi, Laura Sanità

TL;DR

This formulation bypasses hierarchy machinery, offering a more transparent route to the state-of-the-art bound, and 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.

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.

A simple Path-based LP Relaxation for Directed Steiner Tree

TL;DR

This formulation bypasses hierarchy machinery, offering a more transparent route to the state-of-the-art bound, and 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.

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.
Paper Structure (4 sections, 1 theorem, 12 equations)

This paper contains 4 sections, 1 theorem, 12 equations.

Table of Contents

  1. Introduction
  2. Problem Definition and Notations.
  3. LP Relaxation with Path Variables
  4. An easy application of Sherali-Adams Hierarchy

Key Result

Theorem 2.1

The integrality gap of lp:dst for an $\ell$-layered instance of DST is $O(\ell \log k)$.

Theorems & Definitions (2)

  • Theorem 2.1
  • proof