Complexity of Splits Reconstruction for Low-Degree Trees

Abstract

Given a vertex-weighted tree T, the split of an edge xy in T is min{s _x , s _y } where s _x (respectively, s _y ) is the sum of all weights of vertices that are closer to x than to y (respectively, closer to y than to x) in T. Given a set of weighted vertices V and a multiset of splits S, we consider the problem of constructing a tree on V whose splits correspond to S. The problem is known to be NP-complete, even when all vertices have unit weight and the maximum vertex degree of T is required to be no more than 4. We show that

• the problem is strongly NP-complete when T is required to be a path. For this variant we exhibit an algorithm that runs in polynomial time when the number of distinct vertex weights is constant.We also show that

• the problem is NP-complete when all vertices have unit weight and the maximum degree of T is required to be no more than 3, and

• it remains NP-complete when all vertices have unit weight and T is required to be a caterpillar with unbounded hair length and maximum degree at most 3.

Finally, we shortly discuss the problem when the vertex weights are not given but can be freely chosen by an algorithm.

The considered problem is related to building libraries of chemical compounds used for drug design and discovery. In these inverse problems, the goal is to generate chemical compounds having desired structural properties, as there is a strong correlation between structural properties, such as the Wiener index, which is closely connected to the considered problem, and biological activity.

The authors acknowledge the support of Conicyt Chile via projects Fondecyt 11090390 (M.L., K.S.), Fondecyt 11090141 (M.S.), Anillo ACT88 (K.S.), and Basal- CMM (S.G., M.S., K.S.). The first author acknowledges partial support from the European Research Council (COMPLEX REASON, 239962). The second and fourth authors acknowledge the support of the French Agence Nationale de la Recherche (ANR AGAPE ANR-09-BLAN-0159-03).