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

Jordan Dempsey, Leo van Iersel, Mark Jones, Norbert Zeh

TL;DR

It is proved that the integrality gap of the ILP approaches 4 as the size of the two input trees grows, and it is proved 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.

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.

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

TL;DR

It is proved that the integrality gap of the ILP approaches 4 as the size of the two input trees grows, and it is proved that no algorithm based on the ILP formulation by Van Wersch et al. can achieve an approximation ratio of \varepsilon>0$, even on two trees.

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 , for any , 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.
Paper Structure (6 sections, 9 theorems, 3 equations, 1 figure)

This paper contains 6 sections, 9 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An ILP Formulation of the MAF problem
  4. A 4-Approximation Based on LP Rounding
  5. A Family of Tight Inputs
  6. Conclusions

Key Result

Lemma 2.1

Two phylogenetic trees $T_1$ and $T_2$ on the same leaf set $X$ are isomorphic if and only if they have no incompatible quartets.

Figures (1)

  • Figure 1: The two trees $T_1$ and $T_2$ in the proof of \ref{['lem:integral-solution']} for $\ell = 4$.

Theorems & Definitions (15)

  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • Lemma 5.1
  • ...and 5 more