A Flexible, Natural Formulation for the Network Design Problem with Vulnerability Constraints

Given a graph, a set of origin-destination (OD) pairs with communication requirements, and an integer k >= 2, the network design problem with vulnerability constraints (NDPVC) is to identify a subgraph with the minimum total edge costs such that, between each OD pair, there exist a hop-constrained primary path and a hop-constrained backup path after any k - 1 edges of the graph fail. Formulations exist for single-edge failures (i.e., k = 2). To solve the NDPVC for an arbitrary number of edge failures, we develop two natural formulations based on the notion of length-bounded cuts. We compare their strengths and flexibilities in solving the problem for k >= 3. We study different methods to separate infeasible solutions by computing length-bounded cuts of a given size. Experimental results show that, for single-edge failures, our formulation increases the number of solved benchmark instances from 61% (obtained within a two-hour limit by the best published algorithm) to more than 95%, thus increasing the number of solved instances by 1,065. Our formulation also accelerates the solution process for larger hop limits and efficiently solves the NDPVC for general k. We test our best algorithm for two to five simultaneous edge failures and investigate the impact of multiple failures on the network design

A theory of flow network typings and its optimization problems

Many large-scale and safety critical systems can be modeled as flow networks. Traditional approaches for the analysis of flow networks are whole-system approaches in that they require prior knowledge of the entire network before an analysis is undertaken, which can quickly become intractable as the size of network increases. In this thesis we study an alternative approach to the analysis of flow networks, which is modular, incremental and order-oblivious. The formal mechanism for realizing this compositional approach is an appropriately defined theory of network typings. Typings are formalized differently depending on how networks are specified and which of their properties is being verified. We illustrate this approach by considering a particular family of flow networks, called additive flow networks. In additive flow networks, every edge is assigned a constant gain/loss factor which is activated provided a non-zero amount of flow enters that edge. We show that the analysis of additive flow networks, more specifically the max-flow problem, is NP-hard, even when the underlying graph is planar. The theory of network typings gives rise to different forms of graph decomposition problems. We focus on one problem, which we call the graph reassembling problem. Given an abstraction of a flow network as a graph G = (V,E), one possible definition of this problem is specified in two steps: (1) We cut every edge of G into two halves to obtain a collection of |V| one-vertex components, and (2) we splice the two halves of all the edges, one edge at a time, in some order that minimizes the complexity of constructing a typing for G, starting from the typings of its one-vertex components. One optimization is minimizing “maximum” edge-boundary degree of components encountered during the reassembling of G (denoted as α measure). Another is to minimize the “sum” of all edge-boundary degrees encountered during this process (denoted by β measure). Finally, we study different variations of graph reassembling (with respect to minimizing α or β) and their relation with problems such as Linear Arrangement, Routing Tree Embedding, and Tree Layout

A heuristic method for a congested capacitated transit assignment model with strategies

This paper addresses the problem of solving the congested transit assignment problem with strict capacities. The model under consideration is the extension made by Cominetti and Correa (2001), for which the only solution method capable of resolving large transit networks is the one proposed by Cepeda et al. (2006). This transit assignment model was recently formulated by the authors as both a variational inequality problem and a fixed point inclusion problem. As a consequence of these results, this paper proposes an algo- rithm for solving the congested transit assignment problem with strict line capacities. The proposed method consists of using an MSA-based heuristic for finding a solution for the fixed point inclusion formulation. Additionally, it offers the advantage of always obtain- ing capacity-feasible flows with equal computational performance in cases of moderate congestion and with greater computational performance in cases of highly congested net- works. A set of computational tests on realistic small- and large-scale transit networks un- der various congestion levels are reported, and the characteristics of the proposed method are analyzed.Peer ReviewedPostprint (author's final draft