# Improved Inapproximability Results for Steiner Tree via Long Code Based Reductions

## TL;DR

This paper advances the understanding of Steiner tree approximation limits by establishing new inapproximability bounds using Long Code based reductions, connecting the problem to the Unique Games Conjecture, and improving bounds for special graph classes.

## Contribution

It introduces novel Long Code based reductions to establish tighter inapproximability bounds for Steiner tree, including special cases like quasi-bipartite graphs, and relates the problem to the Unique Games Conjecture.

## Key findings

- No polynomial time algorithm can approximate Steiner tree within a factor better than 19/18 unless P=NP.
- It is UG-hard to approximate Steiner tree within a factor better than 17/16.
- In quasi-bipartite graphs, the inapproximability factor is improved to 25/24.

## Abstract

The best algorithm for approximating Steiner tree has performance ratio $\ln(4)+\epsilon \approx 1.386$ [J. Byrka et al., \textit{Proceedings of the 42th Annual ACM Symposium on Theory of Computing (STOC)}, 2010, pp. 583-592], whereas the inapproximability result stays at the factor $\frac{96}{95} \approx 1.0105$ [M. Chleb\'ik and J. Chleb\'ikov\'a, \textit{Proceedings of the 8th Scandinavian Workshop on Algorithm Theory (SWAT)}, 2002, pp. 170-179]. In this article, we take a step forward to bridge this gap and show that there is no polynomial time algorithm approximating Steiner tree with constant ratio better than $\frac{19}{18} \approx 1.0555$ unless \textsf{P = NP}. We also relate the problem to the Unique Games Conjecture by showing that it is \textsf{UG}-hard to find a constant approximation ratio better than $\frac{17}{16} = 1.0625$. In the special case of quasi-bipartite graphs, we prove an inapproximability factor of $\frac{25}{24} \approx 1.0416$ unless \textsf{P = NP}, which improves upon the previous bound of $\frac{128}{127} \approx 1.0078$. The reductions that we present for all the cases are of the same spirit with appropriate modifications. Our main technical contribution is an adaptation of a Set-Cover type reduction in which the Long Code is used to the geometric setting of the problems we consider.

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Source: https://tomesphere.com/paper/1702.02882