Lehman's Theorem and the Directed Steiner Tree Problem
نویسندگان
چکیده
منابع مشابه
On The Integrality Gap of Directed Steiner Tree Problem
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 ac...
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The Directed Steiner Tree (DST) NP-hard problem asks, considering a directed weighted graph with n nodes and m arcs, a node r called root and a set of k nodes X called terminals, for a minimum cost directed tree rooted at r spanning X. The best known polynomial approximation ratio for DST is a O(k)-approximation greedy algorithm. However, a much faster k-approximation, returning the shortest pa...
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Known results: • Generalizes Set Cover, (Non-Metric / Multi-level) Facility Location, Group Steiner Tree • Ω(log2− n)-hard (Halperin, Krauthgamer ’03) • |X| -apx in poly by sophisticated greedy algo (Zelikovsky ’97) • O(log |X|)-apx in nO(log |X|) time by more sophisticated greedy algo (Charikar, Chekuri, Cheung, Goel, Guha and Li ’99) • LP’s have integrality gap Ω( √ k) already for 5 layers; e...
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The goal for the DIRECTED STEINER TREE problem is to find a minimum cost tree in a directed graph G = (V ,E ) that connects all terminals X to a given root r . It is well known that modulo a logarithmic factor it suffices to consider acyclic graphs where the nodes are arranged in l ≤ log |X | levels. Unfortunately the natural LP formulation has a Ω( p |X |) integrality gap already for 5 levels....
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ژورنال
عنوان ژورنال: SIAM Journal on Discrete Mathematics
سال: 2016
ISSN: 0895-4801,1095-7146
DOI: 10.1137/15m1007185