Approximation Algorithms for Flexible Graph Connectivity

We present approximation algorithms for several network design problems in the model of Flexible Graph Connectivity (Adjiashvili, Hommelsheim and M\"uhlenthaler, "Flexible Graph Connectivity", Math. Program. pp. 1-33 (2021), and IPCO 2020: pp. 13-26). Let $k\geq 1$, $p\geq 1$ and $q\geq 0$ be integers. In an instance of the $(p,q)$-Flexible Graph Connectivity problem, denoted $(p,q)$-FGC, we have an undirected connected graph $G = (V,E)$, a partition of $E$ into a set of safe edges $S$ and a set of unsafe edges $U$, and nonnegative costs $c: E\to\Re$ on the edges. A subset $F \subseteq E$ of edges is feasible for the $(p,q)$-FGC problem if for any subset $F'$ of unsafe edges with $|F'|\leq q$, the subgraph $(V, F \setminus F')$ is $p$-edge connected. The algorithmic goal is to find a feasible solution $F$ that minimizes $c(F) = \sum_{e \in F} c_e$. We present a simple $2$-approximation algorithm for the $(1,1)$-FGC problem via a reduction to the minimum-cost rooted $2$-arborescence problem. This improves on the $2.527$-approximation algorithm of Adjiashvili et al. Our $2$-approximation algorithm for the $(1,1)$-FGC problem extends to a $(k+1)$-approximation algorithm for the $(1,k)$-FGC problem. We present a $4$-approximation algorithm for the $(p,1)$-FGC problem, and an $O(q\log|V|)$-approximation algorithm for the $(p,q)$-FGC problem. Finally, we improve on the result of Adjiashvili et al. for the unweighted $(1,1)$-FGC problem by presenting a $16/11$-approximation algorithm. The $(p,q)$-FGC problem is related to the well-known Capacitated $k$-Connected Subgraph problem (denoted Cap-k-ECSS) that arises in the area of Capacitated Network Design. We give a $\min(k,2 u_{max})$-approximation algorithm for the Cap-k-ECSS problem, where $u_{max}$ denotes the maximum capacity of an edge.Comment: 23 pages, 1 figure, preliminary version in the Proceedings of the 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021), December 15-17, (LIPIcs, Volume 213, Article No. 9, pp. 9:1-9:14), see https://doi.org/10.4230/LIPIcs.FSTTCS.2021.9. Related manuscript: arXiv:2102.0330

An oil pipeline design problem

Copyright @ 2003 INFORMSWe consider a given set of offshore platforms and onshore wells producing known (or estimated) amounts of oil to be connected to a port. Connections may take place directly between platforms, well sites, and the port, or may go through connection points at given locations. The configuration of the network and sizes of pipes used must be chosen to minimize construction costs. This problem is expressed as a mixed-integer program, and solved both heuristically by Tabu Search and Variable Neighborhood Search methods and exactly by a branch-and-bound method. Two new types of valid inequalities are introduced. Tests are made with data from the South Gabon oil field and randomly generated problems.The work of the first author was supported by NSERC grant #OGP205041. The work of the second author was supported by FCAR (Fonds pour la Formation des Chercheurs et l’Aide à la Recherche) grant #95-ER-1048, and NSERC grant #GP0105574

The optimal location of facilities on a network

Imperial Users onl

Hierarchical Network Design Using Simulated Annealing

The hierarchical network problem is the problem of nding the least cost net-work, with nodes divided into groups, edges connecting nodes in each groups and groups ordered in a hierarchy. The idea of hierarchical networks comes from telecommunication networks where hierarchies exist. Hierarchical net-works are described and a mathematical model is proposed for a two level version of the hierarchical network problem. The problem is to determine which edges should connect nodes, and how demand is routed in the net-work. The problem is solved heuristically using simulated annealing which as a sub-algorithm uses a construction algorithm to determine edges and route the demand. Performance for dierent versions of the algorithm are reported in terms of runtime and quality of the solutions. The algorithm is able to nd solutions of reasonable quality in approximately 1 hour for networks with 100 nodes