Christoph Moll

Scheduling trams in the morning

Abstract. In this note, we prove ??-completeness of the following problem: Given a set of trams of different types, which are stacked on sidings in their depot and an order in which trams of specified types are supposed to leave. Is there an assignment of trams to departure times without any shunting movements? In the particular case where the number of sidings is fixed, the problem is solvable in polynomial time. We derive a dynamic program and improve its performance by a state elimination scheme. We implemented three variants of the dynamic program and applied them to random data as well as to real-world data.

Tree partitioning under constraints - clustering for vehicle routing problems

Abstract We present a dynamic programming algorithm for the following problem: Given a tree T=(V,E), a set of q non-negative integer weights w i :V→ N on the nodes, and a threshold R i , 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∈T j w i (v)⩽R i 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.