Hypergraphic LP Relaxations for Steiner Trees

We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Könemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. First, we show uncrossing techniques apply to the LP. This implies structural properties, e.g. for any basic feasible solution, the number of positive variables is at most the number of terminals. Second, we show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Third, we give integrality gap upper bounds: an online algorithm gives an upper bound of √ 3 . = 1.729; and in uniformly quasibipartite instances, a greedy algorithm gives an improved upper bound of 73/60 . = 1.216.