A Near-Linear Kernel for Two-Parsimony Distance

The maximum parsimony distance d MP ( T 1 , T 2 ) and the bounded-state maximum parsimony distance d t MP ( T 1 , T 2 ) measure the difference between two phylogenetic trees T 1 , T 2 in terms of the maximum difference between their parsimony scores for any character (with t a bound on the number of states in the character, in the case of d t MP ( T 1 , T 2 ) ). While computing d MP ( T 1 , T 2 ) was previously shown to be fixed-parameter tractable with a linear kernel, no such result was known for d t MP ( T 1 , T 2 ) . In this paper, we prove that computing d t MP ( T 1 , T 2 ) is fixed-parameter tractable for all t . Specifically, we prove that this problem has a kernel of size O ( k lg k ) , where k = d t MP ( T 1 , T 2 ) . As the primary analysis tool, we introduce the concept of leg-disjoint incompatible quartets, which may be of independent interest.