A two-stage network interdiction-monitoring game

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.


INTRODUCTION AND PROBLEM STATEMENT
In this article, we study a two-stage network interdiction-monitoring game involving a defender and an evader. These games take place over a directed network with an origin (source) node and a destination (terminus) node for the evader. In the first stage, the defender interdicts and removes a budget-constrained subset of arcs on the network. We assume that the evader has full knowledge of the interdiction decisions made by the defender. In the second stage, a simultaneous game is played between the defender and the evader: The defender chooses a set of arcs to monitor so as to increase the probability of detection on these arcs, and the evader chooses a path from the source to the terminus with the goal of minimizing the probability of being detected. As modeled by Washburn and Wood [38], this second-stage problem is a two-person simultaneous zero-sum game, that is, both agents reveal their solutions at the same time. The overall objective of the defender is to find an interdiction-monitoring strategy that minimizes the probability that the evader successfully traverses from the source to the terminus without being detected. We refer to this problem as network interdiction in a simultaneous game, or NISG for short.
The NISG could arise when the defender can completely remove arcs in its network in anticipation of an attempt to traverse the network from the source node to the terminus node by the evader. The action of removing the arcs could be expensive, which forces the defender to limit the number of arcs that can be removed, as we do in this article. Monitoring can also be expensive, necessitating the defender's budget limit on monitoring in the second stage. If this monitoring is done surreptitiously, then the second-stage simultaneous game we propose is especially appropriate. In the remainder of this section, we provide a problem statement and illustration in Section 1.1, motivate the problem and conduct a brief literature review in Section 1.2, and summarize the article's contributions in Section 1.3.

Problem statement and illustration
The NISG problem that we study occurs over a directed network G = (N, A), where N is the set of nodes and A is the set of arcs on the network. Letting a be the cost to interdict an arc a ∈ A, the defender modifies the network in the first stage by interdicting (and thus removing) a subset of arcs limited by an interdiction budget of B I . We use x-variables to represent the defender's interdiction decision.
In the second-stage problem, the defender monitors a subset of arcs in A, subject to the given monitoring budget, while the evader seeks to travel from a source node s to a terminus node t undetected. For each arc a ∈ A, let m a be the monitoring cost, let p a ∈ [0, 1) be the probability that the evader is detected on arc a if this arc is not monitored, and let q a ∈ [p a , 1) be this detection probability otherwise. Given a monitoring budget B M , let  = {M ⊆ A ∶ ∑ a∈M m a ≤ B M } be the set of all feasible monitoring decisions, where a feasible monitoring decision corresponds to a subset of arcs that can be monitored at the same time under the monitoring budget. We use y-variables to represent the defender's monitoring strategy, where y M ∈ [0, 1] is the probability that the defender selects a monitoring decision M ∈ . Finally, let  denote the set of all feasible source-terminus paths. We use z-variables to represent the evader's solution, where z P ∈ [0, 1] is the probability that the evader selects path P ∈ . The second-stage simultaneous game involves the evader seeking a solution z that maximizes the probability of reaching the terminus undetected from the source, and the defender seeking a solution y that minimizes this probability. Because both agents are allowed to probabilistically select their solutions in the second stage, their solutions are referred to as mixed strategies. By contrast, if the agents were each obligated to select a single solution (i.e., if the y-and z-variables were restricted to be binary), their solutions are referred to as pure strategies. In simultaneous games such as our second-stage problem, mixed-strategy equilibrium solutions are guaranteed to exist, but pure-strategy equilibrium solutions do not, in general, exist. Additionally, as our second-stage simultaneous game is a finite two-player zero-sum game, each pair of optimal defender (y) and evader (z) solutions yields the same optimal objective value [37]. Thus, the defender is indifferent regarding which optimal monitoring strategy is chosen, if multiple optimal solutions exist.
To illustrate this problem, consider an NISG instance in Figure 1. The evader goes from source node s = 1 to terminus node t = 3. In this instance, we assume a unit cost to interdict and to monitor each arc, that is, a = m a = 1, ∀a ∈ A. For each arc a ∈ A, suppose that the probability that the evader is detected on arc a is p a = 0 if a is not monitored, and is q a otherwise. The q-values are displayed along each arc in Figure 1.  Table 1 shows the impact of each feasible interdiction decision on the probability that the evader is detected while traversing the network. Columns B I , B M , x, Opt monitoring y, Opt path selection z, and Det probability give the interdiction budget, the monitoring budget, the feasible interdictions, the defender's optimal monitoring y-solutions, the evader's optimal path selection z-solutions, and the probability that the evader is detected, respectively. Note that only the nonzero values for the y-and z-variables are reported in Table 1.
Case 1: B I = 0 and B M = 1. The problem is reduced to the simultaneous game previously explored in [38]. The defender's optimal monitoring strategy is to monitor arc (1, 2) with probability 0.385 and arc (1, 3) with probability 0.615. The evader's optimal solution is to select path 1-2-3 with probability 0.385 and path 1-3 with probability 0.615. The probability of the FIGURE 1 NISG instance with a = 1, m a = 1, p a = 0, ∀a ∈ A. The q-values are depicted alongside the arcs.
Not a unique optimal monitoring solution y.
evader being detected at optimality is 0.308. Note that because this is an equilibrium solution, deviating from this monitoring strategy and path selection strategy yields a suboptimal solution. That is, given any other probability distribution that describes the defender's solution, the evader can realize a lower detection probability than 0.308 by changing their path selection probability. Similarly, if the evader's solution changes, then the defender can modify their monitoring probability to yield a detection probability higher than 0.308.  Table 1 shows that in Case 2, each interdiction solution yields a detection probability of 0.5, except for the solution in which arc (1, 3) is interdicted. If arc (1, 3) is removed, then the defender's optimal solution is to monitor arcs (1, 2) and (2, 3) with probability 1, and the evader's optimal strategy is to use path 1-2-3 with probability 1, yielding a detection probability of 0.82. Therefore, the optimal interdiction solution for Case 2 is to remove arc (1, 3).

Motivation and literature review
In network interdiction studies, the defender moves first and modifies the network in some capacity, the impact of which is known by both players. The evader then performs their action, having observed the network information and the impact caused by the defender's decision. Our literature review focuses only on work most directly related to the NISG. For a comprehensive review of progress in this field, we refer the readers to a survey paper by Smith and Song [33].
In classical network interdiction models, the evader's operation corresponds to traversing the network through a shortest path [21] or sending the maximum amount of flow from the source to the sink [39,40]. Fulkerson and Harding [13] study a shortest-path network interdiction problem in which the defender increases the arc length by a continuous amount, subject to a budget limit.
With respect to network interdiction problems in which the defender seeks to minimize the probability that the evader traverses the network undetected, Morton et al. [27] propose nuclear-smuggling interdiction models in which the evader seeks a path having the lowest probability of detection, while the defender maximizes this probability. Lunday and Sherali [24] investigate the linear and nonlinear relationship between applied resources in both covert and overt deployment. Sullivan et al. [35] propose a variant of the problem in which the defender has different and likely more accurate information of the network's parameters than the evader.
The foregoing studies treat these interdiction problems as Stackelberg sequential-play games. For intrusion-detection games, such as the one considered in this article, simultaneous-play models are often used as well. Washburn and Wood [38] study a problem in which the defender monitors an arc on the network with a probability, while the evader probabilistically selects a path on the network to reach a target node. The authors reformulate the problem into a maximum-flow problem and propose a polynomial-time algorithmic solution. Investigating a similar problem, Kodialam and Lakshman [23] propose a monitoring solution that samples subsets of arcs on the network given a budget constraint. Goldberg [15] analyzes a simultaneous nonzero-sum game in which the evader selects a path between a pair of nodes. The defender minimizes the probability that the evader is undetected by allocating inspection resources over the network arcs, where this probability decreases in a nonlinear fashion as the amount of inspection resources increases. The payoffs to the two players differ for each pair of source-sink nodes.
Other studies explore variations of the simultaneous game to reflect different preferences of the defender and the evader. Okamoto et al. [28] examine a network flow interdiction problem in which the evader minimizes inflicted damages, while the defender maximizes damages minus the attack costs. Dahan et al. [11] study a network inspection game in which the defender determines a node inspection strategy that minimizes the number of detectors while achieving a target expected detection rate in the worst case. Basilico et al. [4] consider a multi-stage game with an infinite horizon. The game begins with the defender starting from a node in the network, who then moves to an adjacent node at each subsequent stage. In each stage, the evader can choose to enter the network at a node without knowing the defender's decision, or wait outside the network to observe the defender's strategy. Once entering the network, the evader cannot exit for a specified number of periods, after which the game terminates with two possible outcomes: The defender captures the evader if they share the same node, or the evader successfully leaves the target node.

Contributions of our article
This article contributes a new problem to the network interdiction literature, in which the defender removes arcs in the first stage, and both players participate in a simultaneous game in the second stage via mixed strategies. We provide a formulation to this two-stage network interdiction-monitoring game that allows the adoption of a constraint-and-column generation approach. This becomes the basis for our proposed algorithm to solve the NISG exactly. Lastly, to improve computational efficiency, we introduce enhancements that exploit structures observed in the objective function and feasible interdiction solutions.
The remainder of this article is organized as follows. In Section 2, we introduce a formulation for the NISG, followed by a reformulation that enables the creation of a constraint-and-column generation algorithm for the second-stage problem. We describe this algorithm in Section 3. Section 4 presents a branch-and-cut algorithm that leverages constraint-and-column generation and Benders decomposition to exactly solve the NISG. We provide computational results in Section 5 and finish with concluding remarks in Section 6.

PROBLEM FORMULATION
We start by stating an exact formulation for the NISG in Section 2.1, where the defender's set of feasible monitoring decisions and the evader's set of feasible source-terminus paths are dependent on the (defender's) first-stage interdiction solution. In Section 2.2, we provide a reformulation of NISG where this dependence is removed, which enables the decomposition algorithms that we develop in Section 3.

An exact formulation for the NISG
We first provide formal definitions and additional notation for the NISG. For each arc a ∈ A, define x a as a binary variable such that x a = 1 if the defender removes arc a, and x a = 0 otherwise. Define X = {x ∶ ∑ a∈A a x a ≤ B I , x ∈ {0, 1} |A| } as the set of feasible interdiction decisions that satisfy the interdiction budget constraint. Given a fixed interdiction decision x, define A(x) = {a ∈ A ∶ x a = 0} as the set of remaining arcs on the network, as the set of all feasible monitoring decisions, where a feasible monitoring decision corresponds to a subset of uninterdicted arcs in the network that can be monitored at the same time under the monitoring budget B M , and (x) as the set of feasible source-terminus paths given A(x). Note that (x) ⊆  and (x) ⊆ , ∀x ∈ X. For M ∈  and a ∈ A, define parameter M a = 1 if a ∈ M and M a = 0 otherwise. Using the notation introduced above, we can calculate the probability that the evader is undetected on path P given monitoring decision M as: , ∀P ∈ , M ∈ . Define P 0 = {a 0 } as a dummy single-arc source-terminus path, where a 0 = B I + 1 and F P 0 ,M = 0, ∀M ∈ . That is, arc a 0 cannot be interdicted and the evader is always detected when using path P 0 . Let  ′ (x) = (x) ∪ {P 0 }, ∀x ∈ X. Following the notational convention of Washburn and Wood [38], we formulate the second-stage simultaneous game, given a fixed interdiction decision x ∈ X, as follows: Constraints (1b) and (1d) ensure that the defender's y-solution corresponds to selecting each monitoring decision M ∈ (x) with a nonnegative probability, and the sum of the defender's selection probabilities equals one. Similarly, constraints (1c) and (1e) enforce analogous restrictions with respect to the evader's path selection solution. Note that by introducing P 0 and the associated F-values, we guarantee that if (x) = ∅, then z P 0 = 1 and the optimal objective value equals 0, indicating that it is not possible for the evader to reach the terminus.
We now present a classical linear programming formulation of the simultaneous game by retaining either y or z as variables (we retain y here) and applying a standard epigraph reformulation, where variable u represents the probability that the evader traverses their source-terminus path without being detected. Note that in model (1), the defender's monitoring decision y and the evader's path selection decision z are revealed at the same time. The order of the min and max operators in model (1) is interchangeable in this zero-sum game by applying von Neumann's minimax theorem [37]. This fact can also be observed by deriving the dual of model (2) below.
The objective function minimizes u, which is no less than the maximum probability that the evader is undetected on any path in (x), as indicated by constraints (2b). (Note that if x is chosen so that (x) = ∅, then u = 0 at optimality as desired. Thus, we omit P 0 in (2b).) Therefore, the defender's overall problem-the NISG-is as follows: Observe that for a fixed x, the set of feasible monitoring decisions (x) and the set of feasible paths (x) are finite sets, so there must exist an optimal x-solution that minimizes the probability that the evader is undetected in the simultaneous game. However, since |X|, as well as |(x)| and |(x)| for a fixed x ∈ X, can become exponentially large, we cannot solve this problem directly within practical computational resource limits. Additionally, due to the dependence on x, we cannot utilize constraint-and-column generation with this formulation to avoid the enumeration of (x) and (x), since not every explored path or monitoring decision remains a feasible option for each x ∈ X. This motivates the reformulation proposed in Section 2.2, which becomes the basis for our constraint-and-column generation solution approach for the NISG.

An alternative formulation for the NISG
In this section, we first provide a reformulation of the second-stage problem that removes the dependence of the -sets and -sets on x. We then provide an alternative formulation to the NISG based on this reformulation.
Our proposed reformulation to the second-stage problem is as follows: In this formulation, besides replacing (x) and (x) with  and , respectively, we also introduce the additional x-terms in (4b). For each path P ∈  ⧵ (x), instead of prohibiting the evader from using path P as done in (2b), the right-hand side of constraint (4b) corresponding to P would now be nonpositive. We prove that F-SG(x, , ) is equivalent to model (2) by showing that (2) is equivalent to F-SG(x, , (x)) in Proposition 1, which is then shown to be equivalent to F-SG(x, , ) in Proposition 2.

Proof.
To prove this, we show that for any feasible solution y: (i) The right-hand side of constraint (4b) equals the right-hand side of constraint (2b) for each P ∈ (x); and (ii) The right-hand side of constraint (4b) is nonpositive for each P ∈  ⧵ (x). Since u is nonnegative, at optimality, the objective value obtained from solving (2) must be equal to the objective value obtained from solving F-SG(x, , (x)).
For each arc a ∈ A, the probability that the evader is not detected on arc a is given by 1 − p a − M a (q a − p a ) ∈ (0, 1], since 0 ≤ p a ≤ q a < 1 and M a ∈ {0, 1}, ∀M ∈  by definition. Consider a path P ∈  and a monitoring decision M ∈ (x). The probability that the evader is undetected on P given M is F P,M ∈ (0, 1]. Since any feasible monitoring Given any interdiction decision x, for each unattacked source-terminus path P ∈ (x), we have that ∑ a∈P x a = 0 and constraint (4b) in F-SG(x, , (x)) is identical to (2b). For each source-terminus path P ∈  ⧵ (x) that contains at least one attacked arc, we have ∑ a∈P x a ≥ 1 and . Thus, at optimality, F-SG(x, , (x)) returns a solution y that yields the same optimal objective value obtained by (2). The two models therefore are equivalent. ▪ Proof. To prove the equivalence, we show that there always exists an optimal monitoring solution y , the right-hand side of constraint (4b) corresponding to P remains nonpositive. For each path P ∈ (x), since P ∩ M 1 = P ∩ M 2 , we have: Thus, by setting y ′ M 2 = y M 2 + y M 1 and y ′ M 1 = 0, the value of the right-hand side of constraint (4b) for each P ∈  ⧵ (x) is unaffected. Thus, we still obtain the same optimal objective value u. Since we can repeat this for each monitoring decision in {M ∈  ⧵ (x) ∶ y M > 0} to create an alternative optimal solution to F-SG(x, , ), there must exist an optimal solution (u, By Propositions 1 and 2, we propose the following alternative formulation to solve the NISG: Observe that although F-NISG(, ) can be solved as a mixed-integer program where sets  and  are independent of x, solving it directly by enumerating sets  and  may still be impractical. In Section 3, we propose a model that enables decomposition algorithms to avoid the full enumeration of  and .

The NISG under special considerations
We examine the NISG in two additional settings. The first case involves single-arc monitoring with binary detection probability. We show that when the defender is allowed to monitor only one arc and has perfect detection capability, and the evader always remains undetected on an unmonitored arc, the linear programming formulation provided by [38] can be extended to incorporate interdiction decisions and solved as a classical maximum-flow interdiction problem. The second case considers the NISG under uncertainty, where the defender only has probabilistic information on the detection probabilities and the evader's target node. We show that the solution approach that we present in this article can be readily applied to this setting. More formal problem descriptions and further details of these problems are provided in Appendix C.

A DECOMPOSITION ALGORITHM FOR THE SECOND-STAGE PROBLEM
In this section, we develop a decomposition algorithm based on constraint and column generation to solve the second-stage problem F-SG(x, , ), given a fixed first-stage solutionx. This algorithm will serve as a basis for the branch-and-cut decomposition algorithm for the overall problem F-NISG(, ). The constraint-and-column generation algorithm will start with  ⊆  and  ⊆ , which correspond to a subset of source-terminus paths in the network and a subset of feasible monitoring decisions, respectively. We provide exact mixed-integer programming formulations for the constraint-generation problem and the column-generation problem in Sections 3.1 and 3.2, respectively. We then devise an alternative approach to quickly generate constraints and columns in a heuristic manner in Section 3.3. Section 3.4 integrates these ideas and provides our algorithm for solving the second-stage problem.

Constraint generation
Consider an optimal solution (û,ŷ) to F-SG(x, , ). The constraint-generation problem aims to identify a path P ∈  ⧵  that makes constraint (4b) violated at (û,ŷ), or prove that such a path does not exist. We can do so by identifying a source-terminus path that the defender is least likely to detect the evader using the current monitoring strategy, that is, one that maximizes the right-hand side of constraint (4b) givenŷ. This problem corresponds to a maximum reliability path problem [31].
For each arc (i, j) ∈ A, let binary variable ij represent the evader's traversal decision, where ij = 1 if the evader uses arc (i, j) and ij = 0, otherwise. Let  denote the set of -vectors that satisfy the following classical flow-balance constraints: ∑ j∈N∶(s,j)∈A Let variable f M i denote the maximum possible probability value that the evader is undetected going from node i to terminus t if the defender monitors arcs in M, ∀i ∈ N, M ∈ . The constraint-generation problem can be formulated as the following mixed-integer program: In this formulation, the objective function (8a) maximizes the probability that the evader is undetected starting at the source node for each M ∈ , weighted by the defender's monitoring strategy choiceŷ. Constraints (8b) and (8c) limit the maximum probability that the evader is undetected on an i-t path to be no larger than that on a j-t path multiplied by the probability that the evader is undetected on arc (i, j) if arc (i, j) is used. Constraint (8d) defines f M t to be one, as required, for each M ∈ . Constraint (8e) ensures that the evader only uses unattacked arcs, and constraint (8f) ensures that a source-terminus path is selected.
If the constraint-generation problem RG(x,ŷ, ) is infeasible, then there exists no feasible source-terminus path givenx. Otherwise, if the objective value of RG(x,ŷ, ), which is the maximum probability that the evader reaches terminus t undetected starting from source node s, is larger than û, then the -solution corresponds to a path P ∈  ⧵  that yields a violating constraint (4b) given (û,ŷ).

Column generation
We now consider the column-generation problem to F-SG(x, , ), given an optimal dual solution̂P with respect to constraint (4b), for each P ∈ , and̂with respect to constraint (4c). The column-generation problem aims to identify a new column associated with M ∈  having a negative reduced cost, or prove that such a column does not exist.
Note that the reduced cost corresponding to variable y M is given by Because thêterm is a constant, we focus on minimizing ∑ P∈̂P F P,M below. Let binary variables w represent the monitoring decisions, where w ij = 1 if arc (i, j) is monitored and w ij = 0 otherwise, ∀(i, j) ∈ A. Let variables f P i represent the maximum probability that the evader is undetected going from a node i to terminus t using path P ∈ , given the selected monitoring decision. (Thus, note that f P s plays the role of F P,M , for each P ∈ , where M corresponds to the solution prescribed by the w-variables.) We seek to generate a new column by solving the following formulation: The objective (9a) is a weighted sum of f s -values corresponding to the probability that the evader reaches the terminus undetected. For each path P ∈ , constraints (9b) force the probability that the evader is undetected starting from node i to be no less than the probability of being undetected starting from node j multiplied by the probability of being undetected on arc (i, j) ∈ P, which depends on w ij . Let (f ,ŵ) denote an optimal solution to CG(x,̂, ). If ∑ P∈̂Pf P s −̂< 0, then M = {a ∈ A ∶ŵ a = 1} corresponds to a new column having a negative reduced cost. Otherwise, there exists no negative reduced-cost column.
Observe that (9) is a mixed-integer nonlinear programming formulation due to constraint (9b). To solve the proposed column-generation problem as a mixed-integer (linear) program, we can replace (9b) with the following linear constraints: In summary, both the constraint-generation problem and the column-generation problem can be formulated as mixed-integer programs. However, solving mixed-integer programs iteratively within a decomposition algorithm for the second-stage problem F-SG(x, , ) can be computationally intensive. In fact, both the constraint-generation problem RG(x,ŷ, ) and the column-generation problem CG(x,̂, ) are NP-hard (see Appendix A). This motivates us to develop fast heuristics for constraint and column generation to avoid solving these problems exactly unless necessary.

Constraint and column generation based on linear approximation
In this section, we propose a linear approximation of F P,M -values with respect to arcs, and then use this approximation to obtain heuristic methods for constraint and column generation. The starting point of this linear approximation is to take the logarithm reformulation of the product term associated with F P,M : Applying a linear approximation to the exponential function exp(x) by linear function (1 + x), we define: ) .
As a linear approximation to F P,M , H P,M is an underestimator to F P,M , that is, H P,M ≤ F P,M , and H P,M → F P,M as F P,M → 0, indicating that the quality of this approximation improves if F P,M is closer to 0.
3.3.1 Constraint generation using linear approximation H P,M Similar to the exact constraint generation method in Section 3.1, given an optimal solution (û,ŷ) to F-SG(x, , ), we aim to identify a violating constraint (4b) by maximizing the right-hand side of constraint (4b) givenŷ. To do so, we solve the following optimization problem for constraint generation: where Observe that due to the definition of H P,M in (12), problem (13) can be seen as a shortest-path problem over G = (N, A) where the cost (distance) of each arc a ∈ A is defined as: Note that r a (x) ≥ 0, ∀a ∈ A. The chosen value of R in (13) serves as a "big-M" type parameter, which prohibits the evader from selecting a path that contains an interdicted arc a (i.e.,x a = 1) unless there is no feasible source-terminus path givenx. Let path P ∈  be a shortest path obtained by solving (13) givenx. If a∈Px a > û, then path P corresponds to a violating constraint (4b). If not, however, there may still exist a path P ∈  ⧵  that yields a violating constraint (4b) since H P,M underestimates F P,M , ∀P ∈ , M ∈ . In this case, to determine if there exists a path that violates (4b), we must solve the exact constraint-generation problem using F-values, for example, by using formulation (8). This observation is summarized in the following proposition. (A proof is provided in Appendix B.) Proposition 3. Let (û,ŷ) denote a primal solution obtained by optimizing F-SG(x, , ). Let P ∈  denote a shortest path where the cost of arc a ∈ A is given by r a (x). If P contains an interdicted arc, then no paths in  yield a violating constraint to F-SG(x, , ). If P does not contain an interdicted arc and

3.3.2
Column generation using linear approximation H P,M Similar to the exact column generation method described in Section 3.2, given an optimal dual solution (̂,̂) to F-SG(x, , ), we determine a column that potentially has a negative reduced cost. To avoid solving CG(x,̂, ) unless necessary, we instead use H P,M -values and solve the following problem: Recall that for each a ∈ A and M ∈ , M a = 1 if arc a is in M and M a = 0 otherwise. Based on this definition, problem (15) is a 0-1 knapsack problem (after removing constant terms): Letŵ be an optimal solution to (16), and create set M = {a ∈ A ∶ŵ a = 1}. Note that given solutionŵ, the objective value of (16) plus ∑ P∈̂P minuŝequals the optimal objective value of (15).

A constraint-and-column generation algorithm
We now introduce the SOLVEF-SG function, which uses constraint and column generation to solve the second-stage problem given a first-stage interdiction decisionx. Starting with partial sets  ⊆  and  ⊆ , the SOLVEF-SG function iteratively attempts to identify a violating constraint (the evader's path selection decision) or a negative reduced-cost column (the defender's monitoring decision), and updates partial sets  and  as necessary. Once it is verified that there are no violating constraints or negative reduced-cost columns, the algorithm terminates and reports, with respect tox, the optimal objective value to the second-stage problem û, an optimal monitoring solutionŷ, the corresponding dual solution (̂,̂), and the updated sets  and . Returnû,ŷ,̂,̂, ,  and terminate 21: end if 22: end if 23: end if 24: end if 25: end loop 26: end function Algorithm 1 describes the SOLVEF-SG function. The loop in line 2 iterates until no violating path or negative reduced-cost monitoring decision is identified. Line 3 solves F-SG(x, , ) and obtains a primal solution and a dual solution. We use these solutions to seek violating paths and negative reduced-cost monitoring decisions using the proposed heuristic models in lines 4-10 before applying the exact models in lines 12-18. The order in which we apply each model is as follows.
First, in line 4, we obtain a path P by solving a shortest-path problem where the cost of each arc a ∈ A is given by r a (x), as defined in (14). If line 5 determines that path P corresponds to a violating constraint, then we add P to , and resolve F-SG(x, , ) with the updated . If P does not correspond to a violating constraint, then we proceed to solve (16) to obtain a monitoring decision M in line 8. If line 9 determines that monitoring decision M corresponds to a negative reduced-cost column, then we add M to , and resolve F-SG(x, , ) with the updated  in line 10. Otherwise, we proceed to apply the exact models to solve for additional violating constraints and negative reduced-cost columns, or verify that none exists.
Line 12 obtains a path P for which the evader has the maximum probability of being undetected givenŷ. If line 13 determines that the probability of the evader being undetected on P is larger than û, then we include P in  and add a constraint of the form (4b) to F-SG(x, , ) in line 14 before resolving. Otherwise, line 16 solves CG(x,̂, ) for a most negative reduced-cost monitoring decision M. If M corresponds to a negative reduced-cost column, then in line 18, we include M in  and create a new variable y M in F-SG(x, , ) before resolving. Otherwise, we have verified that there are no violating constraints or negative reduced-cost columns that can be added to F-SG(x, , ). The algorithm then reports primal solution (û,ŷ), dual solution (̂,̂), and partial sets  and , and terminates in line 20. A proof for the following lemma is provided in Appendix B. Lemma 1. Givenx and any subsets  ⊆  and  ⊆ , SolveF-SG(x, , ) generates (updated)  and  such that F-SG(x, , ) and F-SG(x, , ) are equivalent.

A BRANCH-AND-CUT DECOMPOSITION ALGORITHM FOR NISG
In this section, we present a branch-and-cut decomposition algorithm for F-NISG(, ), leveraging the constraint-and-column generation algorithm for the second-stage problem that we developed in Section 3.4. We first demonstrate in Section 4.1 that iteratively applying constraint and column generation alone is insufficient to obtain an optimal x-solution. We then provide in Section 4.2 an epigraph reformulation for the first stage and the corresponding Benders inequalities, followed by an exact branch-and-cut algorithm based on this reformulation. Finally, in Section 4.3, we propose some algorithmic enhancements to the proposed algorithm using supervalid inequalities.

Motivation
Consider the NISG instance depicted in Figure 2. Because B I = 1, the feasible interdiction decisions are to interdict no arc, arc (1, 2), or arc (1, 3). Similarly, the feasible monitoring decisions are to monitor no arc, arc (1, 2), which is denoted as M 1 (corresponding to y 1 ), or arc (1, 3), which is denoted as M 2 (corresponding to y 2 ). The evader can use path 1-2-3, which is denoted as P 1 , or path 1-3, which is denoted as P 2 . (Because the decisions to interdict no arc, or to monitor no arc, are suboptimal, we do not consider them further.) The F-values of all path-monitoring decision pairs are as follows: F P 1 ,M 1 = 0.2, F P 2 ,M 2 = 0.5, and F P 1 ,M 2 = F P 2 ,M 1 = 1. We now demonstrate that given  and  for which at least  ⊂  or  ⊂  is true, we may obtain a suboptimal interdiction solution. Suppose that  = {P 1 , P 2 } and  = {M 2 }. Solving F-NISG(, ) indicates that we should interdict arc (1, 2) and monitor arc (1, 3) with probability one, that is,x 12 = 1 andŷ 2 = 1, yielding an objective value of 0.5. Givenx 12 = 1 and solving F-SG(x, , ), we can determine that no additional paths or monitoring decisions need to be added to  and , respectively. However, the optimal objective value to F-NISG(, ) is 0.2, obtained by interdicting arc (1, 3) (setting x 13 = 1) and monitoring arc (1, 2) with probability one (setting y 1 = 1).
We show that given an interdictionx and partial sets  and ,x is not necessarily optimal even if algorithm SOLVEF-SG returns no new path and no new monitoring decision. We thus propose an epigraph reformulation for the first-stage problem and develop a branch-and-cut algorithm that terminates with an optimal solution to F-NISG(, ).  data ( a , m a , p a , q a ) is shown alongside each arc a ∈ A.

Reformulation and branch-and-cut algorithm
We propose an epigraph mixed-integer programming reformulation to the first-stage problem: This is motivated by the fact that the second-stage value function u(x) is a piecewise-linear convex function. In fact, for any feasible interdictionx, where Λ is the set of dual feasible solutions to F-SG(x, , ). Letting (̂,̂) be any dual solution for F-SG(x, , ), we obtain the following (valid) Benders inequality: Because x-variables are binary and is nonnegative, we can strengthen (18) as follows: where a = min{̂, ∑ P∈∶P∋âP }, ∀a ∈ A. Let  and  denote the set of paths and the set of monitoring decisions obtained when the SOLVEF-SG function terminates, respectively. Note that because F-SG(x, , ) is equivalent to F-SG(x, , ) (by Lemma 1), the dual solution to F-SG(x, , ) also yields a valid inequality for the first-stage problem (17). We use this result to develop Algorithm 2, which is a branch-and-cut algorithm for solving the NISG.
Algorithm 2 begins with partial sets  = {P 0 } and  = {M 0 }, respectively. (In our implementation, we obtain P 0 by solving a shortest-path problem given arc costs r(0) as given by (14). We set M 0 to be a singleton which corresponds to an arbitrary arc a ∈ A that satisfies m a ≤ B M .) We also use ( * , x * , y * ) to track the incumbent solution. The algorithm enters a loop in line 2, which iterates until the optimality of the incumbent solution ( * , x * , y * ) is proven. To verify this, line 3 solves epi-F-NISG(, ) to obtain a first-stage solution (̂,x). Observe that̂and * are a lower bound and an upper bound on the optimal objective value to epi-F-NISG(, ), respectively. Therefore, if line 4 determines that̂= * , then ( * , x * , y * ) is proven to be optimal. (One may instead wish to use a tolerance parameter > 0 and revise line 4 to check that̂≥ * − .) In this case, we report the optimal solution ( * , x * , y * ) and terminate in line 5.
Otherwise, the algorithm proceeds to solve the second-stage problem given interdictionx. Line 6 obtains the optimal second-stage objective value û, an optimal monitoring strategyŷ, dual solution (̂,̂), and updated partial sets  and  by calling the SOLVEF-SG function. Lines 7 and 8 update the incumbent solution as necessary. We add a Benders inequality of the form (19) to epi-F-NISG(, ) in line 9 before resolving.

Supervalid inequalities
In addition to using strengthened Benders inequalities, we apply a class of cuts known as supervalid inequalities (SVIs) to reduce the size of the feasible region, as inspired by Israeli and Wood [21]. Observe that in line 3 of Algorithm 2, the next solution of epi-F-NISG(, ), the interdiction solution cannot improve over the prior solutionx, encountered by the branch-and-cut process, unless an arc that belongs to (x) is interdicted. Accordingly, define A(,x) = ⋃ P∈(x) {a ∈ P} as the union of arcs in feasible source-terminus paths given interdictionx. We propose the following SVI: Add an SVI of the form ∑ a∈A(,x) x a ≥ 1 to epi-F-NISG(, ) 13: end loop In Algorithm 3, we describe a solution approach with SVIs implemented. Algorithm 3 is identical to Algorithm 2 with two exceptions. First, in addition to a Benders inequality, an SVI of the form ∑ a∈A(,x) x a ≥ 1 is also added to epi-F-NISG(, ) (in line 10) before resolving each time. Second, adding an SVI to epi-F-NISG(, ) may cause the model to become infeasible. However, when this occurs, the incumbent solution must be optimal (see Lemma 2). Therefore, besides determining if̂= * , line 4 also checks the feasibility of epi-F-NISG(, ) as a termination condition.

Lemma 2. Constraints (20) do not eliminate an optimal solution to the F-NISG(, ) under Algorithm 3, unless the current incumbent solution x * is optimal.
This is obvious from the fact that for any feasible interdiction solutionx, the evader can always construct a solution by selecting paths from (x). Therefore, we cannot obtain a better objective value to the second-stage problem without attacking an arc belonging to at least one path in (x).

COMPUTATIONAL EXPERIMENTS
We implemented our algorithms in Julia 1.1, using Gurobi 9.0 as the mathematical optimization solver. All computation was done on an HP machine on Clemson University's High Performance Computing Cluster, using a 16-core Intel Xeon E5-2665 with 62GB of RAM. A computational time limit of 3600 s (1 h) was imposed for all instances. Our source code and test instances are available at github.com/diHnguyen/NISG.

Numerical results
In our numerical experiments, we created a total of three data sets, each consisting of 100 randomly generated network instances having |N| nodes and a target network density of 0.1, where |N| = 30, 35, and 40. We set interdiction budget B I = 5 and monitoring budget B M = 10. For each arc a ∈ A, the interdiction cost a and the monitoring cost m a were randomly and independently generated from a discrete uniform distribution ranging from 1 to 10. In our instance generation, we excluded instances in which it is feasible to disconnect the terminus from the source given the instance's B I and -values. Additionally, we randomly generated the detection probability p a if arc a is not monitored, and the detection probability q a if arc a is monitored, from discrete uniform distributions ranging from 0 to 999 and from p a to 999, respectively. We then adjusted p a = p a ∕1000 and q a = q a ∕1000 to obtain p and q-values in [0, 1), for each a ∈ A.
In our implementation, we opt to solve (9) directly as a mixed-integer nonlinear program, rather than using the linearization constraints. Gurobi 9.0 performs the required quadratic linearizations automatically and guarantees that these column-generation problems are solved to optimality. In our preliminary computational experiments, we did not observe a significant computational advantage to solving the linearized version of the problem versus solving (9) directly. This experience will naturally differ depending on the optimization solver employed to solve these problems.
In the remainder of this section, we refer to Algorithm 2 and Algorithm 3 as LP and SVI, respectively. In addition to LP and SVI, we also implemented a third algorithm, which we refer to as IP. Algorithm IP is similar to LP, except that in IP, we always generate new constraints or columns using the exact formulations (i.e., models (8) and (9)) without using any heuristics. The differences between SVI, LP, and IP are summarized in Table 2. Table 3 provides an overview of the performances of our proposed algorithms on the three data sets. Column |N| denotes different instance sizes (via the number of nodes) and column Alg denotes the algorithm implemented. Columns <1 min, O * , and Not solved report the number of instances solved in less than 1 min, number of instances solved between 1 min and 1 h, and number of instances not solved within 1 h, respectively. We observe that the increase in the size of the instance (in terms of the number of nodes) leads to a significant increase in the computational time across all algorithms, as indicated by the increased number of instances not solved within 1 h. Overall, in terms of the number of solved instances within the time limit, IP has the worst performance among the three algorithms, and SVI performs slightly better than LP. Table 4 provides further details on how LP and IP perform compared to SVI. The major column heading SVI reports the computational results using SVI, where columns |N|, O * , and Avg report the number of nodes, the number of instances in each data set solved between 1 min and 1 h (as previously reported in Table 3), and the mean solution time required by SVI to solve the instances that required between 1 min and 1 h to solve.
For the instances reported in column O * under SVI, columns LP and IP report the performance of LP and IP, respectively. Let t SVI and t LP represent the solution time in seconds for an instance using SVI and LP, respectively. Under column LP, column n reports the number of instances reported in O * solved within 1 h using LP. Column Δ t reports the mean difference in solution time over these instances when using LP, where the difference in solution time in each instance is given by t LP −t SVI . Column %Δ t  reports the mean percentage of difference in solution time (also taken over these instances), where the percentage of difference in solution time is given by (t LP − t SVI )∕t SVI × 100%. We report similar results for IP under column IP. SVI tends to solve instances faster than LP or IP. Except for the 30-node instances, where LP performs better than SVI in terms of mean solution time Δ t (but not in terms of %Δ t ), SVI outperforms LP, which significantly outperforms IP. This indicates the effectiveness of the proposed heuristic models that leverage linear approximations to generate new constraints and columns. Additionally, considering SVI and LP, the improvement in solution time (represented by %Δ t and the difference between O * and n) appears to increase as the number of nodes increases, indicating that supervalid inequalities become more important when solving larger network instances.
To further study the efficacy of our proposed branch-and-cut algorithm, Table 5 provides a breakdown of computational time spent on constraint generation and column generation over instances reported in O * by SVI, LP, and IP under column groupings SVI, LP, and IP, respectively. For each of these three groupings, column nRow reports the number of times the corresponding algorithm solves the exact constraint-generation problem, tRow reports the mean percentage of solution time spent on solving the exactly constraint-generation problem, and columns nCol and tCol report analogous data for the column-generation problems.
This table shows that each algorithm spends a small and comparable percentage of computational time solving the column-generation problem tCol, despite the fact that IP requires the solution of more column-generation problems than the other two algorithms nCol. Algorithms SVI and LP require the solution of a comparable number of constraint-generation problems (nRow), while this number is significantly higher for IP. This difference is also reflected in the percentage of computational time (tRow), which is highest in IP. This result further supports our previous findings on the efficacy of the approximate versions of the constraint-and column-generation problems. More specifically, the majority of the solution time in all three algorithms is consistently spent on solving the constraint-generation problem.

Sensitivity analyses on key problem parameters
In this section, we perform several sensitivity analyses on key problem parameters, including the interdiction budget B I and monitoring budget B M , both independently and simultaneously, and the level of detection probability q when arcs are monitored. The experiments in this section were conducted using algorithm SVI due to its superior performance compared to other variants of the proposed branch-and-cut algorithm.
To study the impact of the interdiction budget on solution time, we consider three different levels: B I = 3, 5, and 7. When considering the impact on the optimal objective value of the NISG as the interdiction budget B I varies, we use the network instances that were solved between 1 min and 1 h in the numerical experiments under SVI with B I = 5, and set B I = 1, 3, 5, 7, 9, 11, 13, and 15 on these instances. Similarly, we investigate the impact of the monitoring budget on the solution time by considering three different levels: B M = 7, 10, and 13; we investigate the impact on the objective value by setting B M = 1, 4, 7, 10, 13, 16, 19, and 22. We also look into the impact of varying interdiction budget B I and monitoring budget B M simultaneously on the optimal objective value while setting a common total budget: B I + B M = 15. Specifically, in addition to having (B I , B M ) = (5, 10), for each of the instances solved between 1 min and 1 h under SVI, we created six new settings: (B I , B M ) = (1,14), (3,12), (7,8), (9,6), (11,4), and (13,2). Note that this setting is different from considering the NISG given a shared budget that can be allocated towards either interdiction or monitoring decisions. That is, if the interdiction decision does not exhaust the allocated budget B I , then the remaining budget is discarded, as opposed to being available for use towards monitoring in the second stage.
For the analysis of arc detection probabilities when monitored, that is, q-values, we randomly generated three data sets corresponding to low (L), medium (M), and high (H) level of arc detection probabilities. Specifically, for low level of detection probability, each q a -value is randomly (and independently) generated from a discrete uniform distribution ranging from 400 to 599, and then scaled by q a = q a ∕1000, ∀a ∈ A. Similarly, for medium and high level of detection probabilities, we use discrete uniform distributions that range from 600 to 799 and range from 800 to 999, respectively. We set p a = 0.1 for each arc a ∈ A in this set of experiments.
In Tables 6-8, column |N| reports the number of nodes in the network. Columns B I , B M , and q-level report the interdiction budget, the monitoring budget, and the arc detection probability level when monitored (q-values). We also provide the number     Table 6 reports the impact of varying B I on the computational time. We observe that an increase in the interdiction budget leads to increased solution time. We attribute this behavior to the increased number of feasible interdiction decisions that results from a higher B I -value. Importantly, though, we note that very high values of B I will eventually lead to lower computational times. When B I exceeds the minimum cost of a cut set (i.e., a set of arcs which, when removed, disconnects the terminus from the source), the problem reduces to simply identifying such a minimum-cost cut set, which can be done efficiently. Additionally, if the change in B I leads to a different set of feasible source-terminus paths and feasible monitoring decisions explored in our constraint-and-column generation algorithm, then the solution time is highly dependent on the time spent in solving these subproblems as reported in our numerical results.

Impact of interdiction budget
To study the impact of the interdiction budget on the optimal objective value, we denote by * the optimal objective value of an instance from the original data set (with B I = 5), and bỹthe optimal objective value in the corresponding instance having a modified B I -value. Figure 3 plots the mean and median of the relative difference in the objective value against B I -values, where the relative difference in the objective value in an instance is given by (̃− * )∕ * . (Note that (̃− * )∕ * = 0 when B I = 5.) If  the modified B I -value in an instance is larger than the minimum cost required to disconnect the terminus from the source, then we set̃= 0 without solving the instance using SVI.
As the interdiction budget increases, the mean value of (̃− * )∕ * appears to rapidly approach 0. A larger interdiction budget allows the defender to remove more arcs and monitor the remaining arcs in the network more effectively, leading to a smaller optimal objective value. Additionally, as the B I -value increases, the number of instances in which there exists an x-solution that disconnects the terminus from the source also increases, contributing to the rate at which the mean value of̃-values approaches 0. In particular, half of the reported instances require an interdiction of no larger than 13 units to disconnect the terminus from the source.

5.2.2
Impact of monitoring budget When considering the impact of B I on the objective value, we observed that (̃− * )∕ * approaches 0 rapidly. However, the change measured by (̃− * )∕ * (wherẽnow refers to the revised objective value from choosing a new value of B M ) is less steep compared to when we varied B I . Because p and q-values are randomly generated and are strictly less than one, the probability that the evader is detected is less than one even if the evader uses at least one monitored arc on any source-terminus path. That is, a pure strategy monitoring decision can correspond to an arc set that disconnects the terminus from the source if removed, but monitoring these arcs is less effective in comparison to removing them because the defender cannot detect the evader perfectly. This results in the defender having to monitor more arcs to achieve a similar impact; thus, varying the monitoring budget B M has a more modest impact on (̃− * )∕ * as observed in Figure 4.

5.2.3
Impact of budget allocation We examined the impact of increasing (and decreasing) the interdiction budget B I relative to the monitoring budget B M , while maintaining the same total budget B I + B M . However, these adjustments do not result in any discernible trend in computational time. We provide a detailed discussion of this experiment in Appendix D.1.
Modifying the relative budget allocation also does not have a consistent effect on the objective value. As we increase B I and decrease B M simultaneously by the same number of units, the increase in interdiction budget may not be sufficient to remove an additional arc in one of the source-terminus paths used by the evader, while the decrease in monitoring budget results in a less effective monitoring solution. A similar analysis applies when we decrease B I and increase B M . However, as shown in Figure 5, a general downward trend arises when more budget is dedicated towards interdiction decisions. This aligns with our previous findings in which the allocated interdiction budget exceeds the minimum cost to disconnect the terminus from the source, and changes in monitoring budget has a more modest impact on objective value than changes in interdiction budget. Table 8 reports the computational results when varying the arc detection probability levels (q-values) for 30-node and 35-node instances. (We do not report results on 40-node instances, as none were solved within the time limit.) Overall, we observe that different levels of q-values do not appear to have a clear impact on solution time. However, the number of instances solved within the 1-h time limit decreases significantly in this set of experiments.

Impact of arc detection probability level q-values
One final experiment seeks to determine the effect of modifying q-values on the number of paths generated within our algorithm. The number of paths generated affects computational time required by the algorithm, because our prior experiments showed that solving the constraint-generation problems exactly is the most computationally expensive operation. Details of this experiment are provided in Appendix D.2.

CONCLUSION
In this article, we explored the NISG, a two-stage interdiction problem in which the second stage is a simultaneous game. We provided a formulation to the NISG, and a reformulation that can be solved using a Benders decomposition framework, where the second-stage problem is solved by a constraint-and-column generation algorithm. In addition, we also utilized linear approximation to obtain heuristic constraint-and-column generation and applied supervalid inequalities to improve the computational performance. Our computational experiments showed that the use of linear approximation and supervalid inequalities contributed to improvements in solution time.
Several possible directions exist for future research. From an algorithmic perspective, we could investigate new formulations and solution algorithms for exactly solving the constraint-generation problem (8), since our experimental results indicated that this is a computational bottleneck in the proposed branch-and-cut algorithm. The difficulty in solving these problems is due in large part to the fact that the number of constraints and variables depends on ||, the number of enumerated monitoring sets. Future studies could employ a range of decomposition techniques to elegantly handle the constraint-generation problem.
Additionally, we could examine other variations of the NISG. One such example may arise in situations where the interdiction decisions and monitoring decisions share a common budget. The algorithms given in this article may form a basis for solving such a problem, but key modifications would need to be made to properly extend our proposed approach. In our algorithm, monitoring sets M ∈  remain feasible with respect to the monitoring budget constraint, regardless of the first-stage interdiction. This property would no longer be the case with a shared-budget model. A second example regards an NISG variation in which the first-stage interdiction decision may impact the probability of detection on arcs in the second-stage problem. This situation might arise when the context of the model is a road network. Interdicting an arc might be akin to placing a law enforcement agent on the arc. If the agent detects any evader traversing the arc, then we could interpret that action as akin to removing the arc (since the evader would not use this arc). That agent, however, might be provided some monitoring capability on nearby arcs as well. Hence, interdicting one arc a ∈ A could increase values for p a and/or q a for all arcs that are close to a.
Proof. First, to show that MAXRELPATH belongs to NP, note that a polynomial-size guess can be encoded as a set of arcs. Verifying that the set of arcs forms an s-t path and computing the probability of evasion on this path can clearly be done in polynomial time.
To show that MAXRELPATH is NP-complete, we perform a transformation from 3SAT, which is an NP-complete problem defined as follows [14].
3SAT: Consider a set of n binary variables x 1 , … , x n and k clauses C 1 , … , C k . Each clause C h contains three literals, denoted by c 1h , c 2h , and c 3h , where a literal specifies both a binary variable index and a true or false value. A truth assignment is a choice of true or false values for each binary variable. Clause C h is satisfied by a truth assignment if, for at least one of three literals c gh in the clause, the variable corresponding to c gh takes the value specified by c gh . Does there exist a truth assignment that satisfies every clause?
Our proof transforms any arbitrary 3SAT instance to an equivalent MAXRELPATH instance, that is, the 3SAT instance will have a solution if and only if the transformed MAXRELPATH instance has a solution. This transformation is illustrated in Figure A1. We use the convention in this figure that, for instance, if C 1 specifies that either x 1 is true, or x 2 is false, or x 4 is false, then we represent the clause as C 1 = {x 1 , x 2 , x 4 }. Graph G contains nodes s 1 , … , s n , plus nodes t 1 , … , t k+1 . We create arcs as follows.
• We create two parallel arcs from s i to s i+1 , ∀i ∈ 1, … , n−1: One corresponding to x i and the other corresponding to x i . • Similarly, we create two parallel arcs from s n to t 1 : One corresponding to x n and the other corresponding to x n .
• Finally, we create three parallel arcs from t h and t h+1 , ∀h ∈ 1, … , k: One corresponding to each literal in clause h, as shown in Figure A1.
The source node is s 1 and the sink node is t k+1 . For every arc (i, j) ∈ A (in the MAXRELPATH instance), we set p ij = 0 and q ij = 1.
There are a total of 2n monitoring sets M, each occurring with probabilityŷ M = 1∕(2n). We index the monitoring sets M 1 , … , M 2n , where M i includes all arcs corresponding to setting x i to true, ∀i = 1, … , n, and where M n+i includes all arcs corresponding to setting x i to false, ∀i = 1, … , n. We set the objective target to = 1∕2.
First, suppose that 3SAT has a solution. We create a MAXRELPATH path solution, P, by traversing the arcs exiting the s-nodes that correspond to the 3SAT solution. When leaving node t h , for all h = 1, … , k, we select an outgoing arc that corresponds to a literal that was satisfied by the 3SAT solution. Note that for each i = 1, … , n, path P satisfies exactly one of the following conditions: It traverses at least one arc corresponding to setting x i to true, or it traverses at least one arc corresponding to setting x i to false. Path P therefore traverses arcs covered by exactly n of the 2n monitoring sets. Because the monitoring sets are non-empty, form a partition of A, and are equally likely to occur, the probability of evading detection on path P is 1∕2. This verifies that MAXRELPATH has a solution. Now, suppose that MAXRELPATH has a solution, P. By the above logic, P must not traverse arcs that correspond to more than n different monitoring sets, or else the probability of evasion will be smaller than = 1∕2. However, P traverses exactly one outgoing arc from each s-node, and so P must intersect at least n different monitoring sets (recalling that the arc corresponding to x i that exits s i intersects M i , and the arc corresponding to x i that exits s i intersects M n+i , for all i = 1, … , n). Because P is a solution to MAXRELPATH, when traversing arcs that exit the t-nodes, P must only use those arcs corresponding to literals that the path has traversed when exiting the s-nodes. By doing so, path P must correspond to a truth assignment (gleaned from the literals traversed exiting the s-nodes) that satisfies at least one literal in every clause (verified by the choice of arc taken from each t-node). Hence, 3SAT has a solution as well.
This shows that MAXRELPATH is NP-complete. Because the transformation is made from a strongly NP-complete problem, and only fixed numerical values (0, 1/2, and 1) are used as data in the transformation, MAXRELPATH is also strongly NP-complete. ▪

FIGURE A1
Illustration of the 3SAT transformation to MAXRELPATH, with The column-generation problem reduction Next, recall that the column-generation problem CG(x,̂, ), presented in (9), is designed to find a feasible monitoring set that optimizes the detection probability over a given set of paths. We show that the problem of finding such a monitoring set is NP-hard by establishing the NP-completeness of the following decision problem.
MONITORINGSET: Consider a network G (N, A) having a source node s and sink node t, rational detection probabilities p ij for each (i, j) ∈ A when arcs are not monitored, rational detection probabilities q ij for each (i, j) ∈ A when arcs are monitored (with 0 ≤ p ij ≤ q ij ≤ 1), positive integer arc-monitoring costs m a , ∀a ∈ A, a positive integer arc-monitoring budget B M , and a rational target value . Given a set  of s-t paths, along with rational valueŝP that represent the probability that the evader chooses path P ∈ , does there exist a feasible monitoring set M such that the expected evasion probability given paths in , path-choice probabilitieŝP, and monitoring set M, is no more than ? Proof. To show that MONITORINGSET belongs to NP, note that we can check the feasibility of a guessed solution M to the problem by fixing the w a -values in formulation (9) and solving a linear programming problem in polynomial time over the remaining continuous variables, verifying that the solution is feasible and that the corresponding objective value is no more than the target value.
To show that MONITORINGSET is NP-complete, we perform a transformation from HITTINGSET, which is an NP-complete problem defined as follows [14].
HITTINGSET: Consider a set of elements N = {1, … , n}, and a collection  of subsets of N given by  = {S 1 , … , S m }. Given a positive integer k, does there exist a set H ⊂ N, |H| ≤ k, such that H ∩ S j ≠ ∅, ∀j = 1, … , m?
Our proof transforms any arbitrary HITTINGSET instance to an equivalent MONITORINGSET instance. Graph G for the transformed MONITORINGSET instance contains nodes s 0 , … , s n and t 1 , … , t n+1 , where the source node is s 0 and the sink node is t n+1 . We create a path P j corresponding to each set S j ∈  as follows, where we let 1 , … , l index the elements of S j . In the following steps, if the arc being used by the path does not yet exist in the arc set of graph G, then we create that arc as specified below.
• The path corresponding to S j starts on an arc from s 0 to s 1 with p s 0 s 1 = q s 0 s 1 = 0. • For all i = 1, … , l − 1, the path next uses an arc from s i to t i with p s i t i = 0 and q s i t i = 1, and then an arc from t i to s i+1 with p t i s i+1 = q t i s i+1 = 0. • The path concludes by using an arc from s l to t l with p s l t l = 0 and q s l t l = 1, and an arc from t l to t n+1 with p t l t n+1 = q t l t n+1 = 0.
An illustration of this transformation is provided in Figure A2.
The cost to monitor every arc a ∈ A is given by m a = 1. We choosêP j = 1∕m for each j = 1, … , m. The monitoring budget is B M = k, and the evasion threshold value is = 0.
We now establish equivalence of these instances. First, suppose that there exists a solution H for the HITTINGSET instance. Consider the monitoring set M such that arc (s i , t i ) belongs to M if and only if i ∈ H. No other arcs belong to M. Set M is feasible because it includes only |H| ≤ k arcs. Moreover, by construction of the arc set, every path P ∈  traverses at least one monitored arc of the form (s i , t i ) for some i ∈ {1, … , n}, and is thus detected with probability one. Therefore, the expected probability of evasion is zero, and M is a solution to MONITORINGSET. Now suppose that the MONITORINGSET instance has a solution. Note that a solution always exists in which only arcs of the form (s i , t i ) are monitored (because monitoring any other arc does not improve the probability of detecting the evader on that arc). Let M be one such solution. Define H = {i ∶ (s i , t i ) ∈ M}, and note that |H| ≤ k. Every path P ∈  must traverse at least one monitored arc of the form (s i , t i ), or else it would not guarantee detection of every evader path in . Therefore, H ∩ S j ≠ ∅, ∀j = 1, … , m, and so H is a HITTINGSET solution as well.
Hence, MONITORINGSET is NP-complete, and because only binary numerical data is used in the transformation, MONITORINGSET is strongly NP-complete. ▪ Proof.
If P contains an interdicted arc, then the cost of path P is at least R by definition of r-values. Since P is a shortest path given costs r(x) by assumption, then the cost of any path P ′ ∈  ⧵ {P} is at least R. This implies that P ′ contains at least one interdicted arc and therefore does not yield a violating constraint to F-SG(x, , ) (as a result of Proposition 1).
If P does not contain an interdicted arc, that is,x a = 0, ∀a ∈ P, then we have the following: Proof.
Recall that SOLVEF-SG(x, , ) terminates if there exists no path in  that violates constraint (4b) and if no monitoring decision in  has a negative reduced cost. Let (u ′ , y ′ ) denote an optimal solution to F-SG(x, , ). For each M ∈ , defineỹ M = y ′ M if M ∈  andỹ M = 0 otherwise. We prove that F-SG(x, , ) and F-SG(x, , ) are equivalent by showing that (u ′ ,ỹ) optimizes F-SG(x, , ).
As a result, at termination of the SOLVEF-SG(x, , ) function, we have obtained updated partial sets  and  and an optimal solution to F-SG(x, , ) that is equivalent to F-SG(x, , ), which in turn, is equivalent to F-SG(x, , ). ▪