A Simple 4-Approximation Algorithm for Maximum Agreement Forests on Multiple Unrooted Binary Trees

TL;DR

This paper introduces a straightforward 4-approximation algorithm for the maximum agreement forest problem on multiple unrooted binary trees, using LP rounding on an extended ILP formulation, and proves the tightness of this approximation ratio.

Contribution

It presents a simple LP rounding-based 4-approximation algorithm for multiple unrooted binary trees and establishes the integrality gap limit of the ILP formulation.

Findings

The algorithm achieves a 4-approximation ratio.

The ILP formulation's integrality gap approaches 4.

No ILP-based algorithm can do better than 4-approximate.

Abstract

We present a simple 4-approximation algorithm for computing a maximum agreement forest of multiple unrooted binary trees. This algorithm applies LP rounding to an extension of a recent ILP formulation of the maximum agreement forest problem on two trees by Van Wersch al. We achieve the same approximation ratio as the algorithm of Chen et al. but our algorithm is extremely simple. We also prove that no algorithm based on the ILP formulation by Van Wersch et al. can achieve an approximation ratio of $4 - \varepsilon$, for any $\varepsilon > 0$, even on two trees. To this end, we prove that the integrality gap of the ILP approaches 4 as the size of the two input trees grows.