Linear Programming Hierarchies Suffice for Directed Steiner Tree

We demonstrate that ` rounds of the Sherali-Adams hierarchy and 2` rounds of the Lovász-Schrijver hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in `-layered graphs from Ω( √ k) to O(` · log k) where k is the number of terminals. This is an improvement over Rothvoss’ result that 2` rounds of the considerably stronger Lasserre SDP hierarchy reduce the integrality gap of a similar formulation to O(` · log k). We also observe that Directed Steiner Tree instances with 3 layers of edges have only an O(log k) integrality gap in the standard LP relaxation, complementing the known fact that the gap can be as large as Ω( √ k) in graphs with 4 layers.