# A Linear Programming Based Approach to the Steiner Tree Problem with a Fixed Number of Terminals

## TL;DR

This paper introduces a polynomial-time approach for solving the Steiner tree problem with a fixed number of terminals by using a set of integer programs whose LP relaxations are integral, enabling efficient solutions.

## Contribution

It develops a novel set of polynomial-sized integer programs with integral LP relaxations for the Steiner tree problem with fixed terminals.

## Key findings

- LP relaxations are integral for the proposed IPs
- The approach solves Steiner tree efficiently with fixed terminals
- Number of LPs grows exponentially with terminals, but is manageable for fixed number

## Abstract

We present a set of integer programs (IPs) for the Steiner tree problem with the property that the best solution obtained by solving all, provides an optimal Steiner tree. Each IP is polynomial in the size of the underlying graph and our main result is that the linear programming (LP) relaxation of each IP is integral so that it can be solved as a linear program. However, the number of IPs grows exponentially with the number of terminals in the Steiner tree. As a consequence, we are able to solve the Steiner tree problem by solving a polynomial number of LPs, when the number of terminals is fixed.