The Complexity of Tree Partitioning

Given a tree T on n vertices, and $$k, b, s_1, \ldots , s_b \in \mathbb {N}$$, the Tree Partitioning problem asks if at most k edges can be removed from T so that the resulting components can be grouped into b groups such that the number of vertices in group i is $$s_i$$, for $$i =1, \ldots , b$$. The case where $$s_1=\cdots =s_b =n/b$$, referred to as the Balanced Tree Partitioning problem, was shown to be $${\mathcal {NP}}$$-complete for trees of maximum degree at most 5, and the complexity of the problem for trees of maximum degree 4 and 3 was posed as an open question. The parameterized complexity of Balanced Tree Partitioning was also posed as an open question in another work. In this paper, we answer both open questions negatively. We show that Balanced Tree Partitioning (and hence, Tree Partitioning) is $${\mathcal {NP}}$$-complete for trees of maximum degree 3, thus closing the door on the complexity of Balanced Tree Partitioning, as the simple case when T is a path is in $${\mathcal {P}}$$. In terms of the parameterized complexity of the problems, we show that both Balanced Tree Partitioning and Tree Partitioning are W[1]-complete parameterized by k. Using a compact representation of the solution space for an instance of the problem, we present a dynamic programming algorithm for the weighted version of Tree Partitioning (and hence for that of Balanced Tree Partitioning) that runs in subexponential-time $$2^{O(\sqrt{n})}$$, adding a natural problem to the list of problems that can be solved in subexponential time. Finally, we extend this subexponential-time algorithm to the Weighted Graph Partitioning problem on graphs of treewidth $$o(n/\lg {n})$$, and we also show an application of this subexponential-time algorithm for approximating the Weighted Graph Partitioning problem.