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dc.contributor.authorShadravan, Mohammad 15:35:52 (GMT) 15:35:52 (GMT)
dc.description.abstractIn the Directed Steiner Tree problem, we are given a directed graph G = (V,E) with edge costs, a root vertex r ∈ V, and a terminal set X ⊆ V . The goal is to find the cheapest subset of edges that contains an r-t path for every terminal t ∈ X. The only known polylogarithmic approximations for Directed Steiner Tree run in quasi-polynomial time and the best polynomial time approximations only achieve a guarantee of O(|X|^ε) for any constant ε > 0. Furthermore, the integrality gap of a natural LP relaxation can be as bad as Ω(√|X|).  We demonstrate that l rounds of the Sherali-Adams hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in l-layered graphs from Ω( k) to O(l · log k) 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 · log k). We also observe that Directed Steiner Tree instances with 3 layers of edges have only an O(logk) integrality gap bound in the standard LP relaxation, complementing the fact that the gap can be as large as Ω(√k) in graphs with 4 layers. Finally, we consider quasi-bipartite instances of Directed Steiner Tree meaning no edge in E connects two Steiner nodes V − (X ∪ {r}). By a simple reduction from Set Cover, it is still NP-hard to approximate quasi-bipartite instances within a ratio better than O(log|X|). We present a polynomial-time O(log |X|)-approximation for quasi-bipartite instances of Directed Steiner Tree. Our approach also bounds the integrality gap of the natural LP relaxation by the same quantity. A novel feature of our algorithm is that it is based on the primal-dual framework, which typically does not result in good approximations for network design problems in directed graphs.en
dc.publisherUniversity of Waterlooen
dc.subjectdirected Steiner treeen
dc.subjectapproximation algorithmsen
dc.subjectlift and project methodsen
dc.titleOn the Integrality Gap of Directed Steiner Tree Problemen
dc.typeMaster Thesisen
dc.subject.programCombinatorics and Optimizationen and Optimizationen
uws-etd.degreeMaster of Mathematicsen

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