Abstract
We demonstrate that ℓ rounds of the Sherali-Adams hierarchy and 2ℓ rounds of the Lovász-Schrijver hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in ℓ-layered graphs from \(\Omega(\sqrt k)\) to O(ℓ·logk) where k is the number of terminals. This is an improvement over Rothvoss’ result that 2ℓ rounds of the considerably stronger Lasserre SDP hierarchy reduce the integrality gap of a similar formulation to O(ℓ·logk).
We also observe that Directed Steiner Tree instances with 3 layers of edges have only an O(logk) integrality gap in the standard LP relaxation, complementing the known fact that the gap can be as large as \(\Omega(\sqrt k)\) in graphs with 4 layers.
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Friggstad, Z., Könemann, J., Kun-Ko, Y., Louis, A., Shadravan, M., Tulsiani, M. (2014). Linear Programming Hierarchies Suffice for Directed Steiner Tree. In: Lee, J., Vygen, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2014. Lecture Notes in Computer Science, vol 8494. Springer, Cham. https://doi.org/10.1007/978-3-319-07557-0_24
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DOI: https://doi.org/10.1007/978-3-319-07557-0_24
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