Financial Networks with Intermediation and Transportation Network Equilibria: A Supernetwork Equivalence and Reinterpretation of the Equilibrium Conditions with Computations

In this paper, we consider two distinct classes of network problems – financial networks with intermediation and with electronic transactions and transportation network equilibrium problems, which have been modeled and studied independently. We then prove that the former problem can be reformulated as the latter problem through an appropriately constructed abstract network i.e., a supernetwork. The established equivalence allows one to then transfer the methodological tools, in particular, algorithms, that have been developed for transportation network equilibria to the financial network domain. In addition, this connection provides us with a novel interpretation of the financial network equilibrium conditions in terms of paths and path flows and a direct existence result. We further show how the theoretical results obtained in this paper can be exploited computationally through several numerical examples.      

Some problems in discrete optimization

The present paper concentrates on several problems of network flows and discrete optimization. Progress has been made on some of the problems while little is known about others. Some of the problems discussed are shortest paths, multi-commodity flows, traveling salesman problems, m-center problem, telepak problems and binary trees.This paper was presented at the 7th Mathematical Programming Symposium 1970, The Hague, The Netherlands.Sponsored by the United States Army under Contract No.: DA-31-124-ARO-D-462 and the National Science Foundation, GJ-28339.     

The final answer to the complexity of a basic problem in resilient network design

To provide resilience to failures of the multi-commodity flow network, either in the failure-free state flows can be routed along multiple paths and over-dimensioned, or whenever a failure occurs flows can be restored along unaffected paths. The complexity of the network design depends on the selected method of providing resilience and on a number of design options—whether single or multiple commodities and single- or multi-element failures are considered, if the reaction to failures is dependent or independent on the failure, which mechanism of capacity release and reuse is applied, etc. For almost all combinations of those choices either the corresponding design problem has already been shown to be NP-hard or a compact linear programming formulation of the problem has been provided. The only case that has resisted an answer is when flows are restored in a state-dependent manner using the stub release mechanism. In this paper it is proved that the corresponding network design problem is NP-hard even for a single commodity and for single-element failures. The proof is based on the reduction of the Hamiltonian path problem.     

Increasing Internet Capacity Using Local Search

Open Shortest Path First (OSPF) is one of the most commonly used intra-domain internet routing protocol. Traffic flow is routed along shortest paths, splitting flow evenly at nodes where several outgoing links are on shortest paths to the destination. The weights of the links, and thereby the shortest path routes, can be changed by the network operator. The weights could be set proportional to the physical lengths of the links, but often the main goal is to avoid congestion, i.e. overloading of links, and the standard heuristic recommended by Cisco (a major router vendor) is to make the weight of a link inversely proportional to its capacity.We study the problem of optimizing OSPF weights for a given a set of projected demands so as to avoid congestion. We show this problem is NP-hard, even for approximation, and propose a local search heuristic to solve it. We also provide worst-case results about the performance of OSPF routing vs. an optimal multi-commodity flow routing. Our numerical experiments compare the results obtained with our local search heuristic to the optimal multi-commodity flow routing, as well as simple and commonly used heuristics for setting the weights. Experiments were done with a proposed next-generation AT&T WorldNet backbone as well as synthetic internetworks.     

Dynamic programming algorithms for the elementary shortest path problem with resource constraints

When vehicle routing problems with additional constraints (e.g. capacities or time windows) are solved via column generation and branch-and-price, it is common that the pricing problem requires the computation of a minimum cost constrained path on a graph with costs on the arcs and prizes on the nodes. The pricing problem is usually solved via dynamic programming in two possible ways: requiring elementary paths or allowing paths with cycles. We experimentally compare these two strategies and we evaluate the effectiveness of some algorithmic ideas to improve their performance.     

Routing multi-class traffic flows in the plane

We study a class of multi-commodity flow problems in geometric domains: For a given planar domain P populated with obstacles (holes) of K?2types, compute a set of thick paths from a “source” edge of P to a “sink” edge of P for vehicles of K distinct classes. Each class k of vehicle has a given set, Ok, of obstacles it must avoid and a certain width, wk, of path it requires. The problem is to determine if it is possible to route Nk width-wk paths for class k vehicles from source to sink, with each path avoiding the requisite set Ok of obstacles, and no two paths overlapping. This form of multi-commodity flow in two-dimensional domains arises in computing throughput capacity for multiple classes of aircraft in an airspace impacted by different types of constraints, such as those arising from weather hazards.We give both algorithmic theory results and experimental results.We show hardness of many versions of the problem by proving that two simple variants are NP-hard even in the case K=2. If w₁=w₂=1, then the problem is NP-hard even when O₁=∅. If w₁=2, w₂=3, then the problem is NP-hard even when O₁=O₂. In contrast, the problem for a single width and a single type of obstacles is polynomially solvable.We present approximation algorithms for the multi-criteria optimization problems that arise when trying to maximize the number of routable paths. We also give a polynomial-time algorithm for the case in which the number of holes in the input domain is bounded.Finally, we give experimental results based on an implementation of our methods and experiment with enhanced heuristics for efficient solutions in practice. Our algorithms are being utilized in simulations with NASA?s Future Air traffic management Concepts Evaluation Tool (FACET). We report on experimental results based on applying our algorithms to weather-impacted airspaces, comparing heuristic strategies for searching for feasible path orderings and for computing short multi-class routes. Our results show that multi-class routes can feasibly be computed on real weather data instances on the scale required in air traffic management applications.     

Minimizing maximum risk for fair network connection with interval data

In this paper we introduce a minimax model for network connection problems with interval parameters. We consider how to connect given nodes in a network with a path or a spanning tree under a given budget, where each link is associated with an interval and can be established at a cost of any value in the interval. The quality of an individual link （or the risk of link failure, etc.） depends on its construction cost and associated interval. To achieve fairness of the network connection, our model aims at the minimization of the maximum risk over all links used. We propose two algorithms that find optimal paths and spanning trees in polynomial time, respectively. The polynomial solvability indicates salient difference between our minimax model and the model of robust deviation criterion for network connection with interval data, which gives rise to NP-hard optimization problems.     

Complexity of Bezout’s Theorem VI: Geodesics in the Condition (Number) Metric

We introduce a new complexity measure of a path of (problems, solutions) pairs in terms of the length of the path in the condition metric which we define in the article. The measure gives an upper bound for the number of Newton steps sufficient to approximate the path discretely starting from one end and thus produce an approximate zero for the endpoint. This motivates the study of short paths or geodesics in the condition metric. This work was partly supported by an NSERC Discovery Grant.     

Fractional multi-commodity flow problem: Duality and optimality conditions

This paper deals with multi-commodity flow problem with fractional objective function. The optimality conditions and the duality concepts of this problem are given. For this aim, the fractional linear programming formulation of this problem is considered and the weak duality, the strong direct duality and the weak complementary slackness theorems are proved applying the traditional duality theory of linear programming problems which is different from same results in Chadha and Chadha (2007) [1]. In addition, a strong (strict) complementary slackness theorem is derived which is firstly presented based on the best of our knowledge. These theorems are transformed in order to find the new reduced costs for fractional multi-commodity flow problem. These parameters can be used to construct some algorithms for considered multi-commodity flow problem in a direct manner. Throughout the paper, the boundedness of the primal feasible set is reduced to a weaker assumption about solvability of primal problem which is another contribution of this paper. Finally, a real world application of the fractional multi-commodity flow problem is presented.     

On single-path network routing subject to max-min fair flow allocation

Fair allocation of flows in multicommodity networks has been attracting a growing attention. In Max-Min Fair (MMF) flow allocation, not only the flow of the commodity with the smallest allocation is maximized but also, in turn, the second smallest, the third smallest, and so on. Since the MMF paradigm allows to approximate the TCP flow allocation when the routing paths are given and the flows are elastic, we address the network routing problem where, given a graph with arc capacities and a set of origin-destination pairs with unknown demands, we must route each commodity over a single path so as to maximize the throughput, subject to the constraint that the flows are allocated according to the MMF principle. After discussing two properties of the problem, we describe a column generation based heuristic and report some computational results.     

Graph problems arising from parameter identification of discrete dynamical systems

This paper focuses on combinatorial feasibility and optimization problems that arise in the context of parameter identification of discrete dynamical systems. Given a candidate parametric model for a physical system and a set of experimental observations, the objective of parameter identification is to provide estimates of the parameter values for which the model can reproduce the experiments. To this end, we define a finite graph corresponding to the model, to each arc of which a set of parameters is associated. Paths in this graph are regarded as feasible only if the sets of parameters corresponding to the arcs of the path have nonempty intersection. We study feasibility and optimization problems on such feasible paths, focusing on computational complexity. We show that, under certain restrictions on the sets of parameters, some of the problems become tractable, whereas others are NP-hard. In a similar vein, we define and study some graph problems for experimental design, whose goal is to support the scientist in optimally designing new experiments.     

Graceful reassignment of excessively long communications paths in networks

Modern broadband telecommunications networks transport diverse classes of traffic through flexible end-to-end communications paths. For instance, Internet Protocol (IP) networks with Multi-Protocol Label Switching (MPLS) carry traffic through label switched paths. These flexible paths are often changed in real, or near-real, time in response to congestion and failures detected in the network. As a result, over time, some of these communications paths become excessively long (referred to as out-of-kilter), leading to poor service performance and waste of network resources. An effective reassignment scheme may require reassignment of communications paths with acceptable length (referred to as in-kilter) in order to generate spare capacity on certain links for the out-of-kilter paths. A graceful reassignment solution provides an ordered sequence of reassignments that satisfies the following: (i) the total number of reassigned communications paths does not exceed a specified limit, (ii) no temporary capacity violations are incurred on any network link during the execution of the sequence of reassignments (reassignments are executed sequentially, one at a time), (iii) a communications path is reassigned only as a unit without being split among multiple alternate routes (iv) all reassigned communications paths will be in-kilter, (v) none of the reassignments of communications paths that were originally in-kilter can be excluded from the specified solution without resulting in some capacity violation, and (vi) the sequence of reassignments approximately optimizes a predefined objective, such as maximizing the number of reassigned out-of-kilter communications paths or maximizing the total load reassigned from out-of-kilter communications paths. The resulting problem is formulated as a multi-period, multi-commodity network flow problem with integer variables. We present a search heuristic that takes advantage of certain problem properties to find subsequences of reassignments that become part of the solution, without performing an exhaustive search. Each subsequence reassigns at least one out-of-kilter communication path.     

Improved algorithms for replacement paths problems in restricted graphs

We present near-optimal algorithms for two problems related to finding the replacement paths for edges with respect to shortest paths in sparse graphs. The problems essentially study how the shortest paths change as edges on the path fail, one at a time. Our technique improves the existing bounds for these problems on directed acyclic graphs, planar graphs, and non-planar integer-edge-weighted graphs.     

On the direct path problem of s-elementary frame wavelets

In this paper, we discuss the path-connectivity between two s-elementary normalized tight frame wavelets via the so-called direct paths. We show that the existence of such a direct path is equivalent to the non-existence of an atom of a σ-algebra defined over the defining sets of the corresponding frame wavelets, using a mapping defined by the natural translation and dilation operations between the sets. In particular, this gives an equivalent condition for the existence of a direct path between two s-elementary wavelets.     

New formulations and valid inequalities for a bilevel pricing problem

Consider the problem of maximizing the toll revenue collected on a multi-commodity transportation network. This fits a bilevel framework where a leader sets tolls, while users respond by selecting cheapest paths to their destination. We propose novel formulations of the problem, together with valid inequalities yielding improved algorithms.     

Parameterized complexity of spare capacity allocation and the multicost Steiner subgraph problem

We study the computational complexity of the Spare Capacity Allocation problem arising in optical networks that use a shared mesh restoration scheme. In this problem we are given a network with edge capacities and point-to-point demands, and the goal is to allocate two edge-disjoint paths for each demand (a working path and a so-called restoration path, which is activated only if the working path fails) so that the capacity constraints are satisfied and the total cost of the used and reserved bandwidth is minimized. We focus on the setting where we deal with a group of demands together, and select their restoration paths simultaneously in order to minimize the total cost. We investigate how the computational complexity of this problem is affected by certain parameters, such as the number of restoration paths to be selected, or the treewidth of the network graph. To analyze the complexity of the problem, we introduce a generalization of the Steiner Forest problem that we call Multicost Steiner Subgraph. We study its parameterized complexity, and identify computationally easy and hard cases by providing hardness proofs as well as efficient (fixed-parameter tractable) algorithms.     

A Note on Feasible Flows in Lower-Bounded Multicommodity Networks

This note considers the feasibility for two types of multicommodity flow problems: maximal flow problems with both upper and lower arc capacities, and capacitated minimal cost trans-shipment problems. Although closed form conditions analogous to those known for single commodity problems cannot be derived, it is shown that feasibility is equivalent to finding a maximal multicommodity flow of a specified value on a related network with zero lower bounds, a direct extension of well-known results for single commodity networks.     

A multi-commodity flow formulation for the generalized pooling problem

The pooling problem is an extension of the minimum cost network flow problem where the composition of the flow depends on the sources from which it originates. At each source, the composition is known. In all other nodes, the proportion of any component is given as a weighted average of its proportions in entering flow streams. The weights in this average are simply the arc flow. At the terminals of the network, there are bounds on the relative content of the various components. Such problems have strong relevance in e.g. planning models for oil refining, and in gas transportation models with quality constraints at the reception side. Although the pooling problem has bilinear constraints, much progress in solving a class of instances to global optimality has recently been made. Most of the approaches are however restricted to networks where all directed paths have length at most three, which means that there is no connection between pools. In this work, we generalize one of the most successful formulations of the pooling problem, and propose a multi-commodity flow formulation that makes no assumptions on the network topology. We prove that our formulation has stronger linear relaxation than previously suggested formulations, and demonstrate experimentally that it enables faster computation of the global optimum.     

Multi-period reverse logistics network design

The configuration of the reverse logistics network is a complex problem comprising the determination of the optimal sites and capacities of collection centers, inspection centers, remanufacturing facilities, and/or recycling plants. In this paper, we propose a profit maximization modeling framework for reverse logistics network design problems. We present a mixed-integer linear programming formulation that is flexible to incorporate most of the reverse network structures plausible in practice. In order to consider the possibility of making future adjustments in the network configuration to allow gradual changes in the network structure and in the capacities of the facilities, we consider a multi-period setting. We propose a multi-commodity formulation and use a reverse bill of materials in order to capture component commonality among different products and to have the flexibility to incorporate all plausible means in tackling product returns. The proposed general framework is justified by a case study in the context of reverse logistics network design for washing machines and tumble dryers in Germany. We conduct extensive parametric and scenario analysis to illustrate the potential benefits of using a dynamic model as opposed to its static counterpart, and also to derive a number of managerial insights.