Fixed-parameter tractability for the Tree Assembly problem

Abstract Calculating the “distance” between two given objects with respect to a designated “editing” operation is a hot research area in bioinformatics, where the “distance” is always defined as the minimum number of the “editing” operations required to transform one object into the other one. One of the famous problems in the area is the Minimum Common String Partition problem, which is the simplified variant of the Minimum Tree Cut/Paste Distance problem. Within the paper, we consider another simplified variant of the Minimum Tree Cut/Paste Distance problem, named Tree Assembly problem, of which the edge-deletion operations are specified. More specifically, the Tree Assembly problem aims to transform a given forest into a given tree by edge-addition operations only. In our investigations, we present a fixed-parameter algorithm with runtime 2 O ( k log ⁡ k ) n O ( 1 ) for the Tree Assembly problem, where k is the number of trees in the given forest, and n is the number of nodes in the given tree and forest. Additionally, we give a polynomial time algorithm for a restricted variant of the problem.