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Simplex based Steiner tree instances yield large integrality gaps for the bidirected cut relaxation
The bidirected cut relaxation is the characteristic representative of the
bidirected relaxations () which are a well-known class
of equivalent LP-relaxations for the NP-hard Steiner Tree Problem in Graphs
(STP). Although no general approximation algorithm based on
with an approximation ratio better than for STP is
known, it is mostly preferred in integer programming as an implementation of
STP, since there exists a formulation of compact size, which turns out to be
very effective in practice.
It is known that the integrality gap of is at most
, and a long standing open question is whether the integrality gap is less
than or not. The best lower bound so far is
proven by Byrka et al. [BGRS13]. Based on the work of Chakrabarty et al.
[CDV11] about embedding STP instances into simplices by considering appropriate
dual formulations, we improve on this result by constructing a new class of
instances and showing that their integrality gaps tend at least to .
More precisely, we consider the class of equivalent LP-relaxations
, that can be obtained by strengthening
by already known straightforward Steiner vertex degree
constraints, and show that the worst case ratio regarding the optimum value
between and is at least
. Since is a lower bound for the
hypergraphic relaxations (), another well-known class
of equivalent LP-relaxations on which the current best -approximation algorithm for STP by Byrka et al. [BGRS13] is
based, this worst case ratio also holds for and