Convex hull representation of the deterministic bipartite network interdiction problem

We consider a version of the stochastic network interdiction problem modeled by Morton et al. (IIE Trans 39:3–14, 2007) in which an interdictor attempts to minimize a potential smuggler’s chance of evasion via discrete deployment of sensors on arcs in a bipartite network. The smuggler reacts to sensor deployments by solving a maximum-reliability path problem on the resulting network. In this paper, we develop the (minimal) convex hull representation for the polytope linking the interdictor’s decision variables with the smuggler’s for the case in which the smuggler’s origin and destination are known and interdictions are cardinality-constrained. In the process, we propose an exponential class of easily-separable inequalities that generalize all of those developed so far for the bipartite version of this problem. We show how these cuts may be employed in a cutting-plane fashion when solving the more difficult problem in which the smuggler’s origin and destination are stochastic, and argue that some instances of the stochastic model have facets corresponding to the solution of NP-hard problems. Our computational results show that the cutting planes developed in this paper may strengthen the linear programming relaxation of the stochastic model by as much as 25 %.     

Network design and flow problems with cross-arc costs

Network design and flow problems appear in a wide variety of transportation applications. We consider a new variation to this important class of problems, in which the cost associated with an arc depends not only on the amount of flow moving across that arc, but on the amount of flow on other arcs in the network as well. We formulate an integer program to address this problem, discuss a real-world application in which cross-arc costs are found, and conduct computational experiments on a broad class of problems to analyze how the model performs as network characteristics vary.     

Network deployment of radiation detectors with?physics-based detection probability calculations

We describe a model for deploying radiation detectors on a transportation network consisting of two adversaries: a nuclear-material smuggler and an interdictor. The interdictor first installs the detectors. These installations are transparent to the smuggler, and are made under an uncertain threat scenario, which specifies the smuggler??s origin and destination, the nature of the material being smuggled, the manner in which it is shielded, and the mechanism by which the smuggler selects a route. The interdictor??s goal is to minimize the probability the smuggler evades detection. The performance of the detection equipment depends on the material being sensed, geometric attenuation, shielding, cargo and container type, background, time allotted for sensing and a number of other factors. Using a stochastic radiation transport code (MCNPX), we estimate detection probabilities for a specific set of such parameters, and inform the interdiction model with these estimates.     

A specialized network simplex algorithm for the constrained maximum flow problem

The constrained maximum flow problem is to send the maximum flow from a source to a sink in a directed capacitated network where each arc has a cost and the total cost of the flow cannot exceed a budget. This problem is similar to some variants of classical problems such as the constrained shortest path problem, constrained transportation problem, or constrained assignment problem, all of which have important applications in practice. The constrained maximum flow problem itself has important applications, such as in logistics, telecommunications and computer networks. In this research, we present an efficient specialized network simplex algorithm that significantly outperforms the two widely used LP solvers: CPLEX and lp_solve. We report CPU times of an average of 27 times faster than CPLEX (with its dual simplex algorithm), the closest competitor of our algorithm.     

A Branch and Price algorithm for the k-splittable maximum flow problem

The Maximum Flow Problem with flow width constraints is an NP-hard problem. Two models are proposed: the first model is a compact node-arc model using two flow conservation blocks per path. For each path, one block defines the path while the other one sends the right amount of flow on it. The second model is an extended arc-path model, obtained from the first model after a Dantzig–Wolfe reformulation. It is an extended model as it relies on the set of all the paths between the source and the sink nodes. Some symmetry breaking constraints are used to improve the model. A Branch and Price algorithm is proposed to solve the problem. The column generation procedure reduces to the computation of a shortest path whose cost depends on weights on the arcs and on the path capacity. A polynomial-time algorithm is proposed to solve this subproblem. Computational results are shown on a set of medium-sized instances to show the effectiveness of our approach.     

Shortest path games

We study cooperative games that arise from the problem of finding shortest paths from a specified source to all other nodes in a network. Such networks model, among other things, efficient development of a commuter rail system for a growing metropolitan area. We motivate and define these games and provide reasonable conditions for the corresponding rail application. We show that the core of a shortest path game is nonempty and satisfies the given conditions, but that the Shapley value for these games may lie outside the core. However, we show that the shortest path game is convex for the special case of tree networks, and we provide a simple, polynomial time formula for the Shapley value in this case. In addition, we extend our tree results to the case where users of the network travel to nodes other than the source. Finally, we provide a necessary and sufficient condition for shortest paths to remain optimal in dynamic shortest path games, where nodes are added to the network sequentially over time.     

The quickest path problem with batch constraints

The quickest path problem has been proposed to cope with flow problems through networks whose arcs are characterized by both travel times and flowrate constraints. Basically, it consists in finding a path in a network to transmit a given amount of items from a source node to a sink in as little time as possible, when the transmission time depends on both the traversal times of the arcs and the rates of flow along arcs. This paper is focused on the solution procedure when the items transmission must be partitioned into batches with size limits. For this problem we determine how many batches must be made and what the sizes should be.     

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.     

Network interdiction to minimize the maximum probability of evasion with synergy between applied resources

In this paper, we model and solve the network interdiction problem of minimizing the maximum probability of evasion by an entity traversing a network from a given source to a designated terminus, while incorporating novel forms of superadditive synergy between resources applied to arcs in the network. Inspired primarily by operations to coordinate Iraqi and U.S. security forces seeking to interdict an evader attempting to avoid detection while transiting part of the nearly rectilinear street network in East Baghdad, this study motivates and examines either linear or concave (nonlinear) synergy relationships between the applied resources within our formulations. We also propose an alternative model for sequential overt and covert deployment of subsets of interdiction resources, and conduct theoretical as well as empirical comparative analyses between models for purely overt (with or without synergy) and composite overt-covert strategies to provide insights into absolute and relative threshold criteria for recommended resource utilization. Our empirical results confirm the value of tactical patience regarding decisions on the covert utilization of resources for network interdiction. Furthermore, considering non-integral and integral resource allocations, we identify (theoretically and empirically) parametric characteristics of instances that exhibit the relative worth of employing partially covert operations. Under the relatively more practical scenario involving integral resource allocations, we demonstrate that the composite overt-covert strategy of deploying resources has a greater potential to improve over a purely overt resource deployment strategy, both with and without synergy, particularly when costs are positively correlated, resources are plentiful, and a sufficiently high ratio of covert to overt resources exists. Moreover, should an interdictor be able to ascertain an optimal evader path, the potential and magnitude of this relative improvement for the overt-covert resource allocation strategy is significantly greater.     

Branch-and-price-and-cut algorithms for solving the reliable <Emphasis Type="Italic">h</Emphasis>-paths problem

We examine a routing problem in which network arcs fail according to independent failure probabilities. The reliable h-path routing problem seeks to find a minimum-cost set of h ≥ 2 arc-independent paths from a common origin to a common destination, such that the probability that at least one path remains operational is sufficiently large. For the formulation in which variables are used to represent the amount of flow on each arc, the reliability constraint induces a nonconvex feasible region, even when the integer variable restrictions are relaxed. Prior arc-based models and algorithms tailored for the case in which h = 2 do not extend well to the general h-path problem. Thus, we propose two alternative integer programming formulations for the h-path problem in which the variables correspond to origin-destination paths. Accordingly, we develop two branch-and-price-and-cut algorithms for solving these new formulations, and provide computational results to demonstrate the efficiency of these algorithms.     

Convergent Lagrangian heuristics for nonlinear minimum cost network flows

We consider the separable nonlinear and strictly convex single-commodity network flow problem (SSCNFP). We develop a computational scheme for generating a primal feasible solution from any Lagrangian dual vector; this is referred to as “early primal recovery”. It is motivated by the desire to obtain a primal feasible vector before convergence of a Lagrangian scheme; such a vector is not available from a Lagrangian dual vector unless it is optimal. The scheme is constructed such that if we apply it from a sequence of Lagrangian dual vectors that converge to an optimal one, then the resulting primal (feasible) vectors converge to the unique optimal primal flow vector. It is therefore also a convergent Lagrangian heuristic, akin to those primarily devised within the field of combinatorial optimization but with the contrasting and striking advantage that it is guaranteed to yield a primal optimal solution in the limit. Thereby we also gain access to a new stopping criterion for any Lagrangian dual algorithm for the problem, which is of interest in particular if the SSCNFP arises as a subproblem in a more complex model. We construct instances of convergent Lagrangian heuristics that are based on graph searches within the residual graph, and therefore are efficiently implementable; in particular we consider two shortest path based heuristics that are based on the optimality conditions of the original problem. Numerical experiments report on the relative efficiency and accuracy of the various schemes.     

Optimizing cost flows by edge cost and capacity upgrade

We examine a network upgrade problem for cost flows. A budget can be distributed among the arcs of the network. An investment on each single arc can be used either to decrease the arc flow cost, or to increase the arc capacity, or both. The goal is to maximize the flow through the network while not exceeding bounds on the budget and on the total flow cost.

The problems are NP-hard even on series-parallel graphs. We provide an approximation algorithm on series-parallel graphs which, for arbitrary δ,>0, produces a solution which exceeds the bounds on the budget and the flow cost by factors of at most 1+δ and 1+, respectively, while the amount of flow is at least that of an optimum solution. The running time of the algorithm is polynomial in the input size and 1/(δ). In addition we give an approximation algorithm on general graphs applicable to problem instances with small arc capacities.     

Maximizing the minimum source-sink path subject to a budget constraint

Given a linear cost function for lengthening arcs, a technique is shown for maximizing, within a budget, the shortest source—sink path length in a graph. The computation is equivalent to the parametric solution of a minimum cost flow problem.This work was done while G.C. Harding was at Cornell University.The work of D.R. Fulkerson was supported by the National Science Foundation under Grant MPS74-24026 and by the Office of Naval Research under Grant NR 044-439.     

A direct barter model for course add/drop process

Even though course timetabling and student scheduling problems have been studied extensively, not much has been done for the optimization of student add/drop requests after the initial registration period. Add/drop registrations are usually processed with a first come first served policy. This, however, can introduce inefficiencies and dead-locks resulting in add/drop requests that are not satisfied even though they can, in fact, be satisfied. We model the course add/drop process as a direct bartering problem in which add/drop requests appear as bids. We formulate the resulting problem as an integer linear program. We show that our problem can be solved polynomially as a minimum cost flow network problem. In our model, we also introduce a two-level weighting system that enables students to express priorities among their requests. We demonstrate improvement in the satisfaction of students over the currently used model and also the fast performance of our algorithms on various test cases based on real-life registration data of our university.     

Affirmative action algorithms

Affirmative action is a new variety of selection rule which employs historical information to favor the choice of elements that have not been selected in the past. We categorize three implementations of this principle and discuss their application to the simplex method, to Bard-type schemes for the linear complementarity problem, and to augmenting path methods for network flow problems. We present analytical and computational results, and some open questions.Research supported by the Georgia Institute of Technology.Research supported by the New Faculty Research Development Program of the Georgia Institute of Technology. Reproduction in whole or in part is permitted for any purpose of the U.S. Government.     

Optimal location with equitable loads

The problem considered in this paper is to find p locations for p facilities such that the weights attracted to each facility will be as close as possible to one another. We model this problem as minimizing the maximum among all the total weights attracted to the various facilities. We propose solution procedures for the problem on a network, and for the special cases of the problem on a tree or on a path. The complexity of the problem is analyzed, O(n) algorithms and an O(pn ³) dynamic programming algorithm are proposed for the problem on a path respectively for p=2 and p>2 facilities. Heuristic algorithms (two types of a steepest descent approach and tabu search) are proposed for its solution. Extensive computational results are presented.     

Optimal design of a manpower system

An equilibrium model of a manpower system is developed based on the notion of a career flow. Institutional constraints and measures of system performance are linear functions of the career flow. A typical optimal design problem is formulated and a solution procedure is developed. The optimization problem is a generalized linear program in which columns are generated by solving a shortest path problem. Upper and lower bounds on the optimal value function can be developed at each stage of the calculations.This research was supported by ONR grant N00014-75-C-0619.     

Efficient Parallel Algorithms for the Minimum Cost Flow Problem

In this paper, we propose efficient parallel implementations of the auction/sequential shortest path and the -relaxation algorithms for solving the linear minimum cost flow problem. In the parallel auction algorithm, several augmenting paths can be found simultaneously, each of them starting from a different node with positive surplus. Convergence results of an asynchronous version of the algorithm are also given. For the -relaxation method, there exist already parallel versions implemented on CM-5 and CM-2; our implementation is the first on a shared memory multiprocessor. We have obtained significant speedup values for the algorithms considered; it turns out that our implementations are effective and efficient.     

Virtual path layouts optimizing total hop count on ATM tree networks

This paper presents some algorithmic results concerning virtual path layouts for the one-to-many communication problem in ATM tree networks. The ATM network model is based on covering the network with a layout of virtual paths, under some constraints on the allowed load, namely, the number of paths that can share an edge. The quality measure used is the hop count, namely, the number of edges traversed between two vertices that need to communicate. Whereas most former results concerned the maximum hop count of the virtual path layout, our interest here is in measuring its total hop count, or alternatively its average hop count. The paper presents a dynamic programming algorithm for planning ATM network layouts with minimal total hop count for one-to-many requirements under load constraints over the class of tree networks.