Solving the Maximum Agreement SubTree and the Maximum Compatible Tree problems on many bounded degree trees

TL;DR

This paper investigates the parameterized complexity of the Maximum Agreement SubTree (MAST) and Maximum Compatible Tree (MCT) problems on bounded degree trees, revealing their computational hardness and tight bounds under certain complexity assumptions.

Contribution

It demonstrates that both MAST and MCT are W[1]-hard with respect to maximum degree D and establishes tight lower bounds on their running times, contrasting previous polynomial results for bounded D.

Findings

MAST is W[1]-hard with respect to D.

MAST cannot be solved in O(N^{o(D)}) time unless SNP is in SE.

MCT is W[1]-hard with respect to D and cannot be solved in O(N^{o(2^{D/2})}) time unless SNP is in SE.

Abstract

Given a set of leaf-labeled trees with identical leaf sets, the well-known "Maximum Agreement SubTree" problem (MAST) consists of finding a subtree homeomorphically included in all input trees and with the largest number of leaves. Its variant called "Maximum Compatible Tree" (MCT) is less stringent, as it allows the input trees to be refined. Both problems are of particular interest in computational biology, where trees encountered have often small degrees. In this paper, we study the parameterized complexity of MAST and MCT with respect to the maximum degree, denoted by D, of the input trees. It is known that MAST is polynomial for bounded D. As a counterpart, we show that the problem is W[1]-hard with respect to parameter D. Moreover, relying on recent advances in parameterized complexity we obtain a tight lower bound: while MAST can be solved in O(N^{O(D)}) time where N denotes the input length, we show that an O(N^{o(D)}) bound is not achievable, unless SNP is contained in SE. We also show that MCT is W[1]-hard with respect to D, and that MCT cannot be solved in O(N^{o(2^{D/2})}) time, SNP is contained in SE.