On the Integrality Gap of Directed Steiner Tree Problem

directed Steiner tree;approximation algorithms;lift and project methods;Combinatorics and Optimization

Shadravan, Mohammad

University of Waterloo

关键词: directed Steiner tree;    approximation algorithms;    lift and project methods;    Combinatorics and Optimization;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/8511/5/Shadravan_Mohammad.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

In 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.