Substring Parsimony

Problem Definition

The Substring Parsimony Problem, introduced by Blanchette et al. [1] in the context of motif discovery in biological sequences, can be described in a more general framework:   

Input:

• A discrete space ${ \mathcal{S} }$ on which an integral distance d is defined (i. e. ${ d(x,y) \in \mathbb{N} \ \forall x,y \in \mathcal{S} }$).

• A rooted binary tree ${ T=(V,E) }$ with n leaves. Vertices are labeled ${ \{1,2, \dots,n, \dots,|V|\} }$, where the leaves are vertices ${ \{1,2, \dots,n\} }$.

• Finite sets ${ S_1,S_2, \dots, S_n }$, where set ${ S_i \subseteq \mathcal{S} }$ is assigned to leaf i, for all ${ i=1 \dots n }$.

• A non-negative integer t

Output: All solutions of the form ${ (x_1,x_2, \dots,x_n, \dots,x_{|V|}) }$ such that:

• ${ x_i \in \mathcal{S} }$ for all ${ i =1 \dots |V| }$

• ${ x_i \in S_i }$ for all ${ i=1 \dots n }$

• ${ \sum_{(u,v) \in E} d(x_u,x_v) \leq t }$

The problem thus consists of choosing one element x...