Network design, a cornerstone of mathematical optimization, is about defining the main characteristics of a network satisfying requirements on connectivity, capacity, and level-of-service. It finds applications in logistics and transportation, telecommunications, data sharing, energy distribution, and distributed computing. In multi-commodity network design, one is required to design a network minimizing the installation cost of its arcs and the operational cost to serve a set of point-to-point connections. The definition of this prototypical problem was recently enriched by additional constraints imposing that each origin-destination of a connection is served by a single path satisfying one or more level-of-service requirements, thus defining the Network Design with Service Requirements [Balakrishnan, Li, and Mirchandani. Operations Research, 2017]. These constraints are crucial, e.g., in telecommunications and computer networks, in order to ensure reliable and low-latency communication. In this paper we provide a new formulation for the problem, where variables are associated with paths satisfying the end-to-end service requirements. We present a fast algorithm for enumerating all the exponentially-many feasible paths and, when this is not viable, we provide a column generation scheme that is embedded into a branch-and-cutand-price algorithm. Extensive computational experiments on a large set of instances show that our approach is able to move a step further in the solution of the Network Design with Service Requirements, compared with the current state-of-the-art.
The problem of finding the densest subgraph in a given graph has several applications in graph mining, particularly in areas like social network analysis, protein and gene analyses etc. Depending on the application, finding dense subgraphs can be used to determine regions of high importance, similar characteristics or enhanced interaction. The densest subgraph extraction problem is a fundamentally a non-linear optimization problem. Nevertheless, it can be solved in polynomial time by an exact algorithm based on the iterative solution of a series of maximum flow sub-problems. Despite its polynomial time complexity, the computing time required by the exact algorithms on very large graphs could be prohibitive. Thus, to approach graphs with millions of vertices and edges, one has to resort to heuristic algorithms. We provide an efficient implementation of a greedy heuristic from the literature that is extremely fast and has some nice theoretical properties. We also introduce a new heurisitic algorithm that is built on top of the greedy and the exact methods. An extensive computational study is presented to evaluate the performance of various solution methods on a benchmark composed of 86 instances taken from the literature. This analysis shows that the proposed heuristic algorithm proved very effective on a large number of test instances, often providing either the optimal solution or near-optimal solution within short computing times.
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