On the Optimal Communication Spanning Tree Problem

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Shortest Path versus Multi-Hub Routing in Networks with Uncertain Demand

We study a class of robust network design problems motivated by the need to scale core networks to meet increasingly dynamic capacity demands. Past work has focused on designing the network to support all hose matrices (all matrices not exceeding marginal bounds at the nodes). This model may be too conservative if additional information on traffic patterns is available. Another extreme is the fixed demand model, where one designs the network to support peak point-to-point demands. We introduce a capped hose model to explore a broader range of traffic matrices which includes the above two as special cases. It is known that optimal designs for the hose model are always determined by single-hub routing, and for the fixed- demand model are based on shortest-path routing. We shed light on the wider space of capped hose matrices in order to see which traffic models are more shortest path-like as opposed to hub-like. To address the space in between, we use hierarchical multi-hub routing templates, a generalization of hub and tree routing. In particular, we show that by adding peak capacities into the hose model, the single-hub tree-routing template is no longer cost-effective. This initiates the study of a class of robust network design (RND) problems restricted to these templates. Our empirical analysis is based on a heuristic for this new hierarchical RND problem. We also propose that it is possible to define a routing indicator that accounts for the strengths of the marginals and peak demands and use this information to choose the appropriate routing template. We benchmark our approach against other well-known routing templates, using representative carrier networks and a variety of different capped hose traffic demands, parameterized by the relative importance of their marginals as opposed to their point-to-point peak demands

Designing a road network for hazardous materials shipments

Cataloged from PDF version of article.We consider the problem of designating hazardous materials routes in and through a major population center. Initially, we restrict our attention to a minimally connected network (a tree) where we can predict accurately the flows on the network. We formulate the tree design problem as an integer programming problem with an objective of minimizing the total transport risk. Such design problems of moderate size can be solved using commercial solvers. We then develop a simple construction heuristic to expand the solution of the tree design problem by adding road segments. Such additions provide carriers with routing choices, which usually increase risks but reduce costs. The heuristic adds paths incrementally, which allows local authorities to trade off risk and cost. We use the road network of the city of Ravenna, Italy, to demonstrate the solution of our integer programming model and our path-addition heuristic. © 2005 Elsevier Ltd. All rights reserved

Designing a road network for hazardous materials shipments

We consider the problem of designating hazardous materials routes in and through a major population center. Initially, we restrict our attention to a minimally connected network (a tree) where we can predict accurately the flows on the network. We formulate the tree design problem as an integer programming problem with an objective of minimizing the total transport risk. Such design problems of moderate size can be solved using commercial solvers. We then develop a simple construction heuristic to expand the solution of the tree design problem by adding road segments. Such additions provide carriers with routing choices, which usually increase risks but reduce costs. The heuristic adds paths incrementally, which allows local authorities to trade off risk and cost. We use the road network of the city of Ravenna, Italy, to demonstrate the solution of our integer programming model and our path-addition heuristic. © 2005 Elsevier Ltd. All rights reserved

Solving the Optimum Communication Spanning Tree Problem

This paper presents an algorithm based on Benders decomposition to solve the optimum communication spanning tree problem. The algorithm integrates within a branch-and-cut framework a stronger reformulation of the problem, combinatorial lower bounds, in-tree heuristics, fast separation algorithms, and a tailored branching rule. Computational experiments show solution time savings of up to three orders of magnitude compared to state-of-the-art exact algorithms. In addition, our algorithm is able to prove optimality for five unsolved instances in the literature and four from a new set of larger instances

Dynamic Low-Stretch Trees via Dynamic Low-Diameter Decompositions

Spanning trees of low average stretch on the non-tree edges, as introduced by Alon et al. [SICOMP 1995], are a natural graph-theoretic object. In recent years, they have found significant applications in solvers for symmetric diagonally dominant (SDD) linear systems. In this work, we provide the first dynamic algorithm for maintaining such trees under edge insertions and deletions to the input graph. Our algorithm has update time $n^{1/2 + o(1)}$ and the average stretch of the maintained tree is $n^{o(1)}$, which matches the stretch in the seminal result of Alon et al. Similar to Alon et al., our dynamic low-stretch tree algorithm employs a dynamic hierarchy of low-diameter decompositions (LDDs). As a major building block we use a dynamic LDD that we obtain by adapting the random-shift clustering of Miller et al. [SPAA 2013] to the dynamic setting. The major technical challenge in our approach is to control the propagation of updates within our hierarchy of LDDs: each update to one level of the hierarchy could potentially induce several insertions and deletions to the next level of the hierarchy. We achieve this goal by a sophisticated amortization approach. We believe that the dynamic random-shift clustering might be useful for independent applications. One of these applications is the dynamic spanner problem. By combining the random-shift clustering with the recent spanner construction of Elkin and Neiman [SODA 2017]. We obtain a fully dynamic algorithm for maintaining a spanner of stretch $2k - 1$ and size $O (n^{1 + 1/k} \log{n})$ with amortized update time $O (k \log^2 n)$ for any integer $2 \leq k \leq \log n$. Compared to the state-of-the art in this regime [Baswana et al. TALG '12], we improve upon the size of the spanner and the update time by a factor of $k$.Comment: To be presented at the 51st Annual ACM Symposium on the Theory of Computing (STOC 2019); abstract shortened to respect the arXiv limit of 1920 character

The Optimum Communication Spanning Tree Problem : properties, models and algorithms

For a given cost matrix and a given communication requirement matrix, the OCSTP is defined as finding a spanning tree that minimizes the operational cost of the network. OCST can be used to design of more efficient communication and transportation networks, but appear also, as a subproblem, in hub location and sequence alignment problems. This thesis studies several mixed integer linear optimization formulations of the OCSTP and proposes a new one. Then, an efficient Branch & Cut algorithm derived from the Benders decomposition of one of such formulations is used to successfully solve medium-sized instances of the OCSTP. Additionally, two new combinatorial lower bounds, two new heuristic algorithms and a new family of spanning tree neighborhoods based on the Dandelion Code are presented and tested.Postprint (published version