Integrality Gap of the Hypergraphic Relaxation of Steiner Trees: a short proof of a 1.55 upper bound

TL;DR

This paper provides a shorter proof for the upper bound of 1.55 on the integrality gap of the hypergraphic relaxation used in Steiner tree approximations, enhancing understanding of this relaxation's effectiveness.

Contribution

It offers a concise proof of the 1.55 integrality gap bound by adapting techniques from a recent Steiner tree approximation algorithm.

Findings

Shorter proof of the 1.55 integrality gap bound

Application of randomized loss-contracting algorithm techniques

Improved understanding of hypergraph LP relaxation effectiveness

Abstract

Recently Byrka, Grandoni, Rothvoss and Sanita (at STOC 2010) gave a 1.39-approximation for the Steiner tree problem, using a hypergraph-based linear programming relaxation. They also upper-bounded its integrality gap by 1.55. We describe a shorter proof of the same integrality gap bound, by applying some of their techniques to a randomized loss-contracting algorithm.