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Approximating minimum cost connectivity orientation and augmentation

Singh, Mohit and Végh, László A. ORCID: 0000-0003-1152-200X (2014) Approximating minimum cost connectivity orientation and augmentation. In: SODA 2014 - ACM-SIAM Symposium on Discrete Algorithms, 2014-01-05 - 2014-01-07, Oregon, United States.

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Connectivity augmentation and orientation are two fundamental classes of problems related to graph connectivity. The former includes k-edge-connected subgraph and more generally, survivable network design problem. In orientation problems the goal is to find orientations of an undirected graph that achieve prescribed connectivity properties such as global and rooted k-edge connectivity. In this paper, we consider network design problems that address combined augmentation and orientation settings. We give a polynomial time 6-approximation algorithm for finding a minimum cost subgraph of an undirected graph G that admits an orientation covering a nonnegative crossing G-supermodular demand function, as defined by Frank. The most important example is (k,l)-edge-connectivity, a common generalization of global and rooted edge-connectivity. Our algorithm is based on the iterative rounding method though the application is not standard. First, we observe that the standard linear program is not amenable to the iterative rounding method and therefore use an alternative LP with partition and co-partition constraints. To apply the iterative rounding framework, we first need to introduce new uncrossing techniques to obtain a simple family of constraints that characterize basic feasible solutions. Then we do a careful counting argument based on this family of constraints. We also consider the directed network design problem with orientation constraints where we are given an undirected graph G=(V,E) with costs c(u,v) and c(v,u) for each edge (u,v) in E and an integer k. The goal is to find a subgraph F of minimum cost which has an k-edge connected orientation A. Khanna et al showed that the integrality gap of the natural linear program is at most 4 when k=1 and conjectured that it is constant for all k. We disprove this conjecture by showing an O(|V|)-integrality gap even when k=2.

Item Type: Conference or Workshop Item (Paper)
Additional Information: © 2014 The Authors
Divisions: Mathematics
Subjects: Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Date Deposited: 03 Dec 2014 10:28
Last Modified: 20 Oct 2021 01:08

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