Improved Bi-criteria Approximation for the All-or-Nothing Multicommodity Flow Problem in Arbitrary Networks

This paper addresses the following fundamental maximum throughput routing problem: Given an arbitrary edge-capacitated $n$-node directed network and a set of $k$ commodities, with source-destination pairs $(s_i,t_i)$ and demands $d_i> 0$, admit and route the largest possible number of commodities -- i.e., the maximum {\em throughput} -- to satisfy their demands. The main contributions of this paper are two-fold: First, we present a bi-criteria approximation algorithm for this all-or-nothing multicommodity flow (ANF) problem. Our algorithm is the first to achieve a {\em constant approximation of the maximum throughput} with an {\em edge capacity violation ratio that is at most logarithmic in $n$}, with high probability. Our approach is based on a version of randomized rounding that keeps splittable flows, rather than approximating those via a non-splittable path for each commodity: This allows our approach to work for {\em arbitrary directed edge-capacitated graphs}, unlike most of the prior work on the ANF problem. Our algorithm also works if we consider the weighted throughput, where the benefit gained by fully satisfying the demand for commodity $i$ is determined by a given weight $w_i>0$. Second, we present a derandomization of our algorithm that maintains the same approximation bounds, using novel pessimistic estimators for Bernstein's inequality. In addition, we show how our framework can be adapted to achieve a polylogarithmic fraction of the maximum throughput while maintaining a constant edge capacity violation, if the network capacity is large enough. One important aspect of our randomized and derandomized algorithms is their {\em simplicity}, which lends to efficient implementations in practice

A bi-level programming approach for the shipper-carrier network problem

The Stackelberg game betweenshippers and carriers in an intermodal network is formulated as a bi-levelprogram. In this network, shippers make production, consumption, androuting decisions while carriers make pricing and routing decisions.The oligopolistic carrier pricing and routing problem, which comprisesthe upper level of the bi-level program, is formulated either as a nonlinearconstrained optimization problem or as a variational inequality problem,depending on whether the oligopolistic carriers choose to collude orcompete with each other in their pricing decision. The shippers\u27 decisionbehavior is defined by the spatial price equilibrium principle. Forthe spatial price equilibrium problem, which is the lower level of thebi-level program, a variational inequality formulation is used to accountfor the asymmetric interactions between flows of different commoditytypes. A sensitivity analysis-based heuristic algorithm is proposedto solve the program. An example application of the bi-level programmingapproach analyzes the behavior of two marine terminal operators. Theterminal operators are considered to be under the same Port Authority.The bi-level programming approach is then used to evaluate the PortAuthority\u27s alternative investment strategies

Achieving target equilibria in network routing games without knowing the latency functions

The analysis of network routing games typically assumes precise, detailed information about the latency functions. Such information may, however, be unavailable or difficult to obtain. Moreover, one is often primarily interested in enforcing a desired target flow as an equilibrium. We ask whether one can achieve target flows as equilibria without knowing the underlying latency functions. We give a crisp positive answer to this question. We show that one can efficiently compute edge tolls that induce a given target multicommodity flow in a nonatomic routing game using a polynomial number of queries to an oracle that takes tolls as input and outputs the resulting equilibrium flow. This result is obtained via a novel application of the ellipsoid method, and extends to various other settings. We obtain improved query-complexity bounds for series-parallel networks, and single-commodity routing games with linear latency functions. Our techniques provide new insights into network routing games

Standard Bundle Methods: Untrusted Models and Duality

We review the basic ideas underlying the vast family of algorithms for nonsmooth convex optimization known as "bundle methods|. In a nutshell, these approaches are based on constructing models of the function, but lack of continuity of first-order information implies that these models cannot be trusted, not even close to an optimum. Therefore, many different forms of stabilization have been proposed to try to avoid being led to areas where the model is so inaccurate as to result in almost useless steps. In the development of these methods, duality arguments are useful, if not outright necessary, to better analyze the behaviour of the algorithms. Also, in many relevant applications the function at hand is itself a dual one, so that duality allows to map back algorithmic concepts and results into a "primal space" where they can be exploited; in turn, structure in that space can be exploited to improve the algorithms' behaviour, e.g. by developing better models. We present an updated picture of the many developments around the basic idea along at least three different axes: form of the stabilization, form of the model, and approximate evaluation of the function

Decomposition Methods and Network Design Problems

Decomposition based approaches are recalled from primal and dual point of view. The possibility of building partially disaggregated reduced master problems is investigated. This extends the idea of aggregated-versus-disaggregated formulation to a gradual choice of alternative level of aggregation. Partial aggregation is applied to the linear multicommodity minimum cost flow problem. The possibility of having only partially aggregated bundles opens a wide range of alternatives with different trade-offs between the number of iterations and the required computation for solving it. This trade-off is explored for several sets of instances and the results are compared with the ones obtained by directly solving the natural node-arc formulation. An iterative solution process to the route assignment problem is proposed, based on the well-known Frank Wolfe algorithm. In order to provide a first feasible solution to the Frank Wolfe algorithm, a linear multicommodity min-cost flow problem is solved to optimality by using the decomposition techniques mentioned above. Solutions of this problem are useful for network orientation and design, especially in relation with public transportation systems as the Personal Rapid Transit. A single-commodity robust network design problem is addressed. In this, an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. A set of new instances that are computationally hard for the natural flow formulation are solved by means of a new heuristic algorithm. Finally, an efficient decomposition-based heuristic approach for a large scale stochastic unit commitment problem is presented. The addressed real-world stochastic problem employs at its core a deterministic unit commitment planning model developed by the California Independent System Operator (ISO)

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

An Examination of Railroad Capacity and its Implications for Rail-Highway Intermodal Transportation

After many years of decline in market share, railroads are now experiencing an increasing demand for their services. Service intensive intermodal transportation seems to be an especially promising market area. Since the historic decline in traffic has been accompanied by a reduction in network infrastructure, however, the railroads\u27 ability to handle sizable traffic increases, at least in the short term, is in question. Since rail transportation is critical to the domestic economy of the nation, and is increasingly important in international logistics channels, shortfalls in railroad capacity are not desirable. The published literature on railroad capacity is relatively sparse, especially in comparison to the highway mode. Much of what is available pertains to individual network components such as lines or terminals. Evaluation of system capacity, considering the interactive effects of traffic flowing through a network of lines and terminals, has received less attention. A tool specifically designed for evaluating freight railroad system capacity issues could be a useful addition to the rail analyst\u27s toolbox. The research conducted in this study resulted in the formulation and application of RAILNET, a multicomrnodity, multicarrier network model for predicting equilibrium flows within a railroad network. Designed for strategic planning with a short term horizon, the model assumes fixed external demand. The predicted flows meet the conditions for Wardropian system equilibrium. At completion, the solution algorithm predicts the expected delay per train on each link, allowing the analyst to identify areas of congestion. Following completion of the model, it was applied to a case study examining the railroad network in the southeastern U.S. The public use version of the Interstate Commerce Commission\u27s Commodity Waybill Sample (CWS) provided flow data. The dissertation describes the procedure used to develop the case study and presents some results. The case points to major deficiencies in the CWS data which resulted in substantially less traffic in the network than is actually present. In general, given this limitation, the model behaved well and results appear reasonable, although not necessarily reflective of actual network conditions

Achieving Target Equilibria in Network Routing Games without Knowing the Latency Functions

The analysis of network routing games typically assumes, right at the onset, precise and detailed information about the latency functions. Such information may, however, be unavailable or difficult to obtain. Moreover, one is often primarily interested in enforcing a desirable target flow as the equilibrium by suitably influencing player behavior in the routing game. We ask whether one can achieve target flows as equilibria without knowing the underlying latency functions. Our main result gives a crisp positive answer to this question. We show that, under fairly general settings, one can efficiently compute edge tolls that induce a given target multicommodity flow in a nonatomic routing game using a polynomial number of queries to an oracle that takes candidate tolls as input and returns the resulting equilibrium flow. This result is obtained via a novel application of the ellipsoid method, and applies to arbitrary multicommodity settings and non-linear latency functions. Our algorithm extends easily to many other settings, such as (i) when certain edges cannot be tolled or there is an upper bound on the total toll paid by a user, and (ii) general nonatomic congestion games. We obtain tighter bounds on the query complexity for series-parallel networks, and single-commodity routing games with linear latency functions, and complement these with a query-complexity lower bound applicable even to single-commodity routing games on parallel-link graphs with linear latency functions. We also explore the use of Stackelberg routing to achieve target equilibria and obtain strong positive results for series-parallel graphs. Our results build upon various new techniques that we develop pertaining to the computation of, and connections between, different notions of approximate equilibrium, properties of multicommodity flows and tolls in series-parallel graphs, and sensitivity of equilibrium flow with respect to tolls. Our results demonstrate that one can indeed circumvent the potentially-onerous task of modeling latency functions, and yet obtain meaningful results for the underlying routing game