Simplex based Steiner tree instances yield large integrality gaps for the bidirected cut relaxation
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The bidirected cut relaxation is the characteristic representative of the bidirected relaxations (BCR) 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 BCR with an approximation ratio better than 2 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 BCR is at most 2, and a long standing open question is whether the integrality gap is less than 2 or not. The best lower bound so far is 36/31≈ 1.161 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 6/5 = 1.2. More precisely, we consider the class of equivalent LP-relaxations BCR^+, that can be obtained by strengthening BCR by already known straightforward Steiner vertex degree constraints, and show that the worst case ratio regarding the optimum value between BCR and BCR^+ is at least 6/5. Since BCR^+ is a lower bound for the hypergraphic relaxations (HYP), another well-known class of equivalent LP-relaxations on which the current best (ln(4) + ε)-approximation algorithm for STP by Byrka et al. [BGRS13] is based, this worst case ratio also holds for BCR and HYP.
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