On the Integrality Gap of the Directed-Component Relaxation for Steiner Tree

TL;DR

This paper proves that the integrality gap of the $k$-Directed-Component Relaxation LP for the Steiner tree problem is at most approximately 1.39, providing an efficient method to find near-optimal Steiner trees.

Contribution

It establishes a tight upper bound on the integrality gap of the $k$-DCR LP and offers a constructive, efficient approach to approximate Steiner trees within this factor.

Findings

Integrality gap of $k$-DCR LP is at most $oxed{ ext{ln}(4)} ext{ } extless 1.39$.

Constructive method to find Steiner trees within the integrality gap.

Efficient algorithm for near-optimal Steiner tree approximation.

Abstract

In this note, we show that the integrality gap of the $k$-Directed-Component- Relaxation($k$-DCR) LP for the Steiner tree problem, introduced by Byrka, Grandoni, Rothvob and Sanita (STOC 2010), is at most $\ln(4)<1.39$. The proof is constructive: we can efficiently find a Steiner tree whose cost is at most $\ln(4)$ times the cost of the optimal fractional $k$-restricted Steiner tree given by the $k$-DCR LP.