Tree partitioning under constraints — clustering for vehicle routing problems Original Research Article

We present a dynamic programming algorithm for the following problem: Given a tree T=(V,E), a set of q non-negative integer weights wi:V→N on the nodes, and a threshold Ri, i=1,…,q. Partition the vertices of the tree into connected components T0,…,Tk, such that for all i∈{1,…,q}, j∈{0,…,k} ∑v∈Tjwi(v)⩽Ri and k is minimal. We show that this problem is hard, if q is unbounded or if T has unbounded maximum degree. In all other cases the running time of the dynamic program has a polynomial worst-case bound.