Abstract We present a reinforcement learning‐based heuristic for a two‐player interdiction game called the dynamic shortest path interdiction problem (DSPI). The DSPI involves an evader and an interdictor who take turns in the problem, with the interdictor selecting a set of arcs to attack and the evader choosing an arc to traverse at each step of the game. Our model employs the Monte Carlo tree search framework to learn a policy for the players using randomized roll‐outs. This policy is stored as an asymmetric game tree and can be further refined as the game unfolds. We leverage alpha–beta pruning and existing bounding schemes in the literature to prune suboptimal branches. Our numerical experiments demonstrate that the prescribed approach yields near‐optimal solutions in many cases and allows for flexibility in balancing solution quality and computational effort.

Recent studies employ collections of binary decision diagrams (BDDs) to solve combinatorial optimization problems. This paper focuses on the problem of optimally aligning two BDDs, that is, transforming them to enforce a common order of variables while keeping the total size of the diagrams as small as possible. We address this problem, which is known to be NP-hard, by introducing and studying a simplified problem instead of working with the more complex original diagrams. We discuss some basic properties of the simplified problem, design a corresponding heuristic for the original problem, and show empirically that this approach yields good quality alignments while significantly reducing the complexity of intermediate diagram transformations. We highlight the practicality of this approach in the context of a variation of the uncapacitated facility location problem. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms–Discrete. Funding: This work was supported by the Office of Naval Research [Grant N00014-17-1-2421]. Supplemental Material: The online appendices are available at https://doi.org/10.1287/ijoc.2023.1293 .

Abstract We study a network interdiction problem involving two agents: a defender and an evader. The evader seeks to traverse a path from a source node to a terminus node in a directed network without being detected. The game takes place in two stages. In the first stage, the defender removes a set of arcs in the network. In the second stage, the defender and evader play a simultaneous game. The defender monitors a set of arcs, thus increasing the probability that the evader will be detected on that arc (if the evader uses the arc). The evader selects a source‐terminus path. Because the second stage is played simultaneously, both agents use mixed‐strategy solutions. We approach the solution of the second‐stage problem by proposing a constraint‐and‐column generation algorithm. We show that both the constraint‐generation and column‐generation problems are NP‐hard. Accordingly, we prescribe approximate versions of these problems that can be solved more efficiently. Our algorithm relies on solving the approximate versions until it is necessary to obtain an exact solution of the constraint‐generation and column‐generation problems. Then, to link the first‐ and second‐stage problems, we model the original problem using an epigraph reformulation, which we solve using a Benders‐decomposition based approach. The efficacy of our approach is demonstrated on a set of randomly generated test instances.

Abstract We consider a class of maximum flow problems (MFPs) having node‐ and arc‐capacity constraints, in which each unit of flow on arc consumes a positive amount of capacity at nodei. This problem arises in wireless sensor network optimization applications, where node capacities refer to sensor energy limits. This version of the MFP is traditionally solved using linear programming due to the presence of the complicating node‐capacity constraints. As an alternative scheme, we prescribe an approach for this problem based on augmenting flows along paths and cycles, showing why sending flows on augmenting cycles becomes necessary in this class of problems. Although our augmenting flow algorithm ultimately requires the solution of auxiliary linear programs, we demonstrate the computational advantages of our approach on randomly generated test instances.

This paper examines a class of three-stage sequential defender-attacker-defender problems. In these problems the defender first selects a subset of assets to protect, the attacker next damages a subset of unprotected assets in the “interdiction” stage, after which the defender optimizes a “recourse” problem over the surviving assets. These problems are notoriously difficult to optimize, and almost always require the recourse problem to be a convex optimization problem. Our contribution is a new approach to solving defender-attacker-defender problems. We require all variables in the first two stages to be binary-valued, but allow the recourse problem to take any form. The proposed framework focuses on solving the interdiction problem by restricting the defender to select a recourse decision from a sample of feasible vectors. The algorithm then iteratively refines the sample to force finite convergence to an optimal solution. We demonstrate that our algorithm not only solves interdiction problems involving NP-hard recourse problems within reasonable computational limits, but it also solves shortest path fortification and interdiction problems more efficiently than state-of-the-art algorithms tailored for that problem, finding optimal solutions to real-road networks having up to 300,000 nodes and over 1,000,000 arcs.

We consider graph search problems involving an intruder and mobile searchers. The graph consists of nodes on which the intruder and searchers may be located, and edges on which these entities travel. Associated with each node is a set of nodes that are visible from that node. The goal is to find the minimum number of searchers needed to detect the intruder within a given time limit. We investigate three variants of the graph search problem: (i) a hide‐and‐seek problem, in which a stationary intruder “hides” at an unknown node, (ii) a pursuit‐evasion problem, in which the intruder moves among the nodes to avoid being detected, and (iii) a patrol problem, which is similar to the pursuit‐evasion problem except that searchers patrol the graph in repeated circuits to seek intruders. Our contribution provides exponential‐size set‐covering formulations for these problems, along with a class of branch‐cut‐price algorithms tailored for solving them. These algorithms leverage results from the orienteering literature to solve pricing problems related to searcher routes. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(1), 4–18 2017

We study a dynamic network game between an attacker and a user. The user wishes to find a shortest path between a pair of nodes in a directed network, and the attacker seeks to interdict a subset of arcs to maximize the user's shortest‐path cost. In contrast to most previous studies, the attacker can interdict arcs any time the user reaches a node in the network, and the user can respond by dynamically altering its chosen path. We assume that the attacker can interdict a limited number of arcs, and that an interdicted arc can still be traversed by the user at an increased cost. The challenge is therefore to find an optimal path (possibly repeating arcs in the network), coupled with the attacker's optimal interdiction strategy (i.e., which arcs to interdict and when to interdict them). We propose an exact exponential‐state dynamic‐programming algorithm for this problem, which can be reduced to a polynomial‐time algorithm in the case of acyclic networks. We also develop lower and upper bounds on the optimal objective function value based on classical interdiction and robust optimization models, or based on an exact solution to variations of this problem. We examine the efficiency of our algorithms and the quality of our bounds on a set of randomly generated instances. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(4), 315–330 2016

Consider a directed network in which each arc can fail with some specified probability. An entity arrives on this network at a designated origin node and traverses the network in a random‐walk fashion until it either terminates at a destination node, or until an arc fails while being traversed. We study the problem of assessing the probability that the random walk reaches the destination node, which we call the survival probability of the network. Complicating our analysis is the assumption that certain arcs have “memory,” in the sense that after a memory arc is successfully traversed, it cannot fail on any subsequent traversal during the walk. We prove that this problem is #P‐hard, provide methods for obtaining lower and upper bounds on the survival probability, and demonstrate the effectiveness of our bounding methods on randomly generated networks. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 66(1), 67–86 2015

We address a variant of the Euclidean traveling salesman problem known as the close-enough traveling salesman problem (CETSP), where the traveler visits a node if it enters a compact neighborhood set of that node. We formulate a mixed-integer programming model based on a discretization scheme for the problem. Both lower and upper bounds on the optimal CETSP tour length can be derived from the solution of this model, and the quality of the bounds obtained depends on the granularity of the discretization scheme. Our approach first develops valid inequalities that enhance the bound and solvability of this formulation. We then provide two alternative formulations, one that yields an improved lower bound on the optimal CETSP tour length, and one that greatly improves the solvability of the original formulation by recasting it as a two-stage problem amenable to decomposition. Computational results demonstrate the effectiveness of the proposed methods.

We consider an interdiction problem that involves an operator (or defender) whose goal is to maximize flow from a source node to a sink node in some network that resides in Euclidean space. The problem we examine takes the perspective of an interdictor, who seeks to minimize the defender's maximum flow by locating a set of attacks that diminish arc capacities in accordance with the distance from the arc to the attack. Attacks are not restricted to node or arc locations, and can occur anywhere on the region in which the network is located. We refer to this problem as the Euclidean maximum flow network interdiction problem (E‐MFNIP). We show that E‐MFNIP is NP‐hard, as it generalizes the maximum flow interdiction problem studied by Wood . This article contributes two approaches to solving E‐MFNIP based on solving a sequence of lower‐bounding integer programs from which upper bounds can be readily obtained, and shows that these bounds are convergent. Computations on a set of test instances indicate that an approach based on space‐discretization tends to converge much faster than one based on linearizing the nonlinear capacity functions. We demonstrate the application of our space‐discretization approach on a real geographical network. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 64(2), 109–124 2014

Abstract This article considers the problem of using synchronous mobile agents to decontaminate the nodes of a graph given a spreading contamination. We begin by considering the problem of minimizing cleaning time, given initial agent, and contamination locations. Then, we take as input a set of all possible locations in which a contamination can start and examine problems in which we strategically preposition agents. In one problem, we minimize the number of agents and prescribe their initial locations, so that the graph can be cleaned within a time limit for any potential initial contamination. We also determine the best initial locations for some predetermined number of agents to minimize expected cleaning time, given probability estimates of potential initial contamination locations. We analyze the complexity of each variant and formulate the problems as mixed‐integer programs. As an alternative method, we also provide a construction heuristic for the cleaning problem and cutting‐plane algorithms for the agent location problems. Computational results using these approaches demonstrate the efficacy of our procedures. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

We propose an algorithm for maximizing the lifetime of a wireless sensor network when there is a mobile sink and the underlying application can tolerate some amount of delay in delivering the data to the sink. The algorithm is distributed, and in addition, mostly uses local information. Such an algorithm can be implemented by parallel and/or distributed execution and the overhead of message passing is low. It is also possible to embed the algorithm into a network protocol so that the sensor nodes and the sink can run it directly as part of the network operation. We give a proof of the algorithm's optimality and the boundedness of the queue sizes, both in the long-run average sense. The proof is based on analyzing a Lyapunov drift.

Abstract We examine variants of the critical node problem on specially structured graphs, which aim to identify a subset of nodes whose removal will maximally disconnect the graph. These problems lie at the intersection of network interdiction and graph theory research and are relevant to several practical optimization problems. The two different connectivity metrics that we consider regard the number of maximal connected components (which we attempt to maximize) and the largest component size (which we attempt to minimize). We develop optimal polynomial‐time dynamic programming algorithms for solving these problems on tree structures and on series‐parallel graphs, corresponding to each graph‐connectivity metric. We also extend our discussion by considering node deletion costs, node weights, and solving the problems on generalizations of tree structures. Finally, we demonstrate the computational efficacy of our approach on randomly generated graph instances. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012

We consider a network where nodes communicate by exchanging information packets whose fields include the address of the sending node and that of the destination node. In the absence of some verification mechanism, an attacking node can send packets to another node using a forged origin address. In this study, we consider an optimization problem of identifying a minimum cardinality subset of verification nodes on a tree such that the number of attacks from any forged origin to any destination is limited to a prescribed level. For the case in which communication is permitted between every node in the tree, we develop an optimal polynomial-time dynamic programming algorithm for this problem. We compare the performance of the dynamic programming algorithm against a mixed-integer programming model on randomly generated tree networks at varied levels of security.

Abstract In this article, we examine the design of an evacuation tree, in which evacuation is subject to capacity restrictions on arcs. The cost of evacuating people in the network is determined by the sum of penalties incurred on arcs on which they travel, where penalties are determined according to a nondecreasing function of time. Given a discrete set of disaster scenarios affecting network population, arc capacities, transit times, and penalty functions, we seek to establish an optimal a priori evacuation tree that minimizes the expected evacuation penalty. The solution strategy is based on Benders decomposition, in which the master problem is a mixed‐integer program and each subproblem is a time‐expanded network flow problem. We provide efficient methods for obtaining primal and dual subproblem solutions, and analyze techniques for improving the strength of the master problem formulation, thus reducing the number of master problem solutions required for the algorithm's convergence. We provide computational results to compare the efficiency of our methods on a set of randomly generated test instances. © 2008 Wiley Periodicals, Inc. NETWORKS, 2009

Most traditional routing problems assume perfect operability of all arcs and nodes. However, when independent arc failure probabilities exist, a secondary objective must be present to retain some measure of expected functionality, introducing nonlinear, nonconvex constraints. We examine the robust two-path problem, which seeks to establish two paths between a source and destination node wherein at least one path must remain fully operable with some threshold probability. We consider the case where both paths must be arc disjoint and the case where arcs can be shared between the paths. We examine various strategies for solving the resulting nonlinear integer program, including pruning, coefficient tightening, lifting, and branch-and-bound partitioning schemes. We discuss the advantages and disadvantages of these methods and conclude with computational results.

This paper introduces and studies an optimization problem related to the alias method for discrete random-variate generation. The alias method is an efficient method to generate random variates from a discrete probability distribution. The efficiency of the alias method can be improved by designing the alias table such that the expected number of computations that must be performed per value generated is minimized. The problem of optimizing the construction of the alias table is proven to be strongly NP-hard, even if either of two variations of the alias method relaxing the alias-table-generation restrictions are used. Integer-programming formulations describing these three optimization problems are presented, and insights regarding necessary optimality criteria and relationships among their optimal solutions are discussed.

The Conditional Covering Problem (CCP) is a facility location problem on a graph, where the set of nodes represents demand points and potential facility locations. The key aspect of the CCP is that each facility covers all nodes within a given facility-specific coverage radius, except for the node at which it is located. The objective of this problem is to minimize the sum of the facility location costs required to cover all demand points. We first discuss the worst-case complexity of the CCP by examining literature related to the total domination problem, which is a special case of the CCP. Next, we examine the special case of path graphs and provide an O(n2) algorithm for its solution. Finally, we leverage information obtained from this procedure to derive an optimal algorithm for “extended star” graphs (multiple paths having one node in common), without increasing the worst-case complexity of the algorithm. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(4), 177–185 2005

In a previous article, we presented algorithms for solving the Conditional Covering Problem (CCP) on path and extended star graphs. The CCP on these graphs can be solved in O(n2) time, where n is the number of nodes in the graph. In this article, we propose a new dynamic programming procedure to solve the CCP on tree graphs. This recursion works from the leaf nodes of the tree up to the root node, using notions of protected and unprotected costs as done for the CCP path algorithm in our previous work. We introduce new preliminary routines and data structures to merge information from subpaths and subtrees, resulting in an O(n4) algorithm to optimally solve the problem. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(4), 186–197 2005

Abstract We develop stochastic integer programming techniques tailored toward solving a Synchronous Optical Network (SONET) ring design problem with uncertain demands. Our approach is based on an L‐shaped algorithm, whose (integer) master program prescribes a candidate network design, and whose (continuous) subproblems relay information regarding potential shortage penalty costs to the ring design decisions. This naive implementation performs very poorly due to two major problems: (1) the weakness of the master problem relaxations, and (2) the limited information passed to the master problem by the optimality cuts. Accordingly, we enforce certain necessary conditions regarding shortage penalty contributions to the objective function within the master problem, along with a corresponding set of valid inequalities that improves the solvability of the master problem. We also show how a nonlinear reformulation of the model can be used to capture an exponential number of optimality cuts generated by the linear model. We augment these techniques with a powerful upper‐bounding heuristic to further accelerate the convergence of the algorithm, and demonstrate the effectiveness of our methodologies on a test bed of randomly generated stochastic SONET instances. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(1), 12–26 2004

In this paper, we consider a network design problem arising in the context of deploying synchronous optical networks (SONET) using a unidirectional path switched ring architecture, a standard of transmission using optical fiber technology. Given several rings of this type, the problem is to find an assignment of nodes to possibly multiple rings, and to determine what portion of demand traffic between node pairs spanned by each ring should be allocated to that ring. The constraints require that the demand traffic between each node pair should be satisfiable given the ring capacities, and that no more than a specified maximum number of nodes should be assigned to each ring. The objective function is to minimize the total number of node-to-ring assignments, and hence, the capital investment in add-drop multiplexer equipments. We formulate the problem as a mixed-integer programming model, and propose several alternative modeling techniques designed to improve the mathematical representation of this problem. We then develop various classes of valid inequalities for the problem along with suitable separation procedures for tightening the representation of the model, and accordingly, prescribe an algorithmic approach that coordinates tailored routines with a commercial solver (CPLEX). We also propose a heuristic procedure which enhances the solvability of the problem and provides bounds within 5–13% of the optimal solution. Promising computational results are presented that exhibit the viability of the overall approach and that lend insights into various modeling and algorithmic constructs.