# An Improved Integrality Gap for Steiner Tree

## TL;DR

This paper presents a combinatorial primal-dual algorithm that significantly improves the upper bound on the integrality gap of the bidirected cut relaxation for Steiner tree problems from 2 to 1.5, bringing it closer to the conjectured lower bound.

## Contribution

It introduces a new combinatorial algorithm that tightens the upper bound on the integrality gap for Steiner tree relaxations, advancing theoretical understanding.

## Key findings

- Upper bound on integrality gap improved to 3/2
- Algorithm based on primal-dual schema
- Brings upper bound closer to the conjectured lower bound

## Abstract

A promising approach for obtaining improved approximation algorithms for Steiner tree is to use the bidirected cut relaxation (BCR). The integrality gap of this relaxation is at least $36/31$, and it has long been conjectured that its true value is very close to this lower bound. However, the best upper bound for general graphs was an almost trivial $2$. We improve this bound to $3/2$ by a combinatorial algorithm based on the primal-dual schema.