# 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_MP^t(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_MP^t(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_MP^t(T_1,T_2). In this paper, we prove that computing d_MP^t(T_1, T_2) is fixed-parameter tractable for all t. Specifically, we prove that this problem has a kernel of size O(k k), where k = d_MP^t(T_1, T_2). As the primary analysis tool, we introduce the concept of leg-disjoint incompatible quartets, which may be of independent interest.

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