The maximum parsimony distance $d_{\textrm{MP}}(T_1,T_2)$ and the bounded-state maximum parsimony distance $d_{\textrm{MP}}^t(T_1,T_2)$ 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 $d_{\textrm{MP}}^t(T_1,T_2)$). While computing $d_{\textrm{MP}}(T_1, T_2)$ was previously shown to be fixed-parameter tractable with a linear kernel, no such result was known for $d_{\textrm{MP}}^t(T_1,T_2)$. In this paper, we prove that computing $d_{\textrm{MP}}^t(T_1, T_2)$ is fixed-parameter tractable for all~$t$. Specifically, we prove that this problem has a kernel of size $O(k \lg k)$, where $k = d_{\textrm{MP}}^t(T_1, T_2)$. As the primary analysis tool, we introduce the concept of leg-disjoint incompatible quartets, which may be of independent interest.
翻译:$d ⁇ textrm{MP}}(T_1,T_2美元) 最大保值距离(T_1,T_2美元) 和约束状态最大保值最大保值距离($d ⁇ trm{MP}}(T_1,T_2美元) $d{t_t_1,T_2美元) 。计算美元最大保值距离(T_t_t_t_1,T_2美元) 最大保值距离(T_t_t_t_1美元) 最大保值距离(T_t_tlextrmrm{MP}t$(T_t_1,T_2美元) 最大保值距离(T_t_t_1,T_T_2美元) 最大保值(t_t_tr_2美元) 。在计算公式数(t_1,t_1,T_T_T_T_2美元) 和最大保值(T_K_r_r_r_r_r_r__r_x_x_x_x_x_x_x_x_T_x_x_xx_x_x_x_x_x_x_xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx