05. Models for Railway Track Allocation

Abstract

The optimal track allocation problem (OPTRA) is to find, in a given
railway network, a conflict free set of train routes of maximum
value. We study two types of integer programming formulations for
this problem: a standard formulation that models block conflicts in
terms of packing constraints, and a novel formulation of the
`extended' type that is based on additional `configuration'
variables. The packing constraints in the standard formulation stem
from an interval graph and can therefore be separated in polynomial
time. It follows that the LP-relaxation of a strong version of this
model, including all clique inequalities from block conflicts, can
be solved in polynomial time.  We prove that the LP-relaxation of
the extended formulation can also be solved in polynomial time, and
that it produces the same LP-bound. Albeit the two formulations are
in this sense equivalent, the extended formulation has advantages
from a computational point of view. It features a constant number of
rows and is amenable to standard column generation
techniques. Results of an empirical model comparison on mesoscopic
data for the Hanover-Fulda-Kassel region of the German long
distance railway network are reported.