Introduction 131

The shortest path (SP) problem has been called “the simplest of all network flow problems” (Ahuja et al., 1993, p. 6). Fulkerson and Harding (1977) introduced the idea of attacking segments of a network to the SP problem by maximizing the length of a minimum shortest path. Golden (1978) furthered Fulkerson’s work and used the term

network interdiction problem. Wood (1993) showed that the shortest path network

interdiction problem is NP-complete. Improvements to the algorithm for solving the shortest path network interdiction problem were described by Israeli and Wood (2002).

The simple SP problem becomes the much more difficult constrained shortest path (CSP) problem when an additional budgetary side constraint is added, and the special network structure of the original problem can no longer be exploited (Ahuja et al., 1993, p. 599). Our research expands the complexity of the network interdiction problem by adding two new features. First, we expand the network interdiction problem to model constrained shortest paths. Second, we introduce the ability to defend segments of the network against attack. Even though the literature starts with and focuses on network flow problems, the algorithms in Alderson et al. (2011, 2014, & 2015) also apply to very general operator models, including those modeled as integer linear programming (ILP) problems. We picked CSP as the operator problem because it is a well understood specimen of an ILP that is almost a network flow problem. The addition of defenses to the network interdiction problem along with the addition of a side constraint creates the Defender-Attacker -Defender Constrained Shortest Path (DAD CSP) problem.

The concept behind the DAD CSP model is similar to the general DAD model presented in Chapter I. For DAD CSP, our objective is to travel from source to destination at a minimum cost, and the additional budgetary constraint specifies that the total time to travel from source to destination must be within a time budget. The attacker desires to interdict some of the arcs of the network graph, limited by an attack budget.

Each attack increases the cost of traversing an arc by a fixed amount. The defender has the ability to prevent attacks on a subset of arcs in the network, based on a given defense budget. We do not consider adding new arcs as defenses in this chapter. Thus, the DAD CSP problem seeks to choose defenses to minimize the cost of the constrained shorted path resulting from a cost maximizing worst-case attack.

We define the DAD CSP problem as follows. Given a network G = (N,A) with a prescribed start node s, destination node t, arcs with costs cij and travel times tij, and an

overall time budget, T, the classical CSP problem seeks the least cost path from s to t such that the time expended on the path is no larger than T. The choice of arcs on the constrained shortest path is governed by binary variables (Y). This is the simplest version of the operator problem (4.1a), which we seek to embellish. If we add a distance penalty, qij, to each arc so that the length of an attacked arc (X) becomes cij + qij, we obtain the

attacker problem (4.1b). If we add the ability to protect a subset of arcs from an attack

(W) constrained by a defense budget, we obtain the defender problem (4.1c). The constraint sets for defenses (Ψ), attacks (Ξ) and paths (ϒ) in (4.1a)-(4.1c) are similar to those in Chapter I, and are suppressed for clarity. The operator model constraint set in (4.1c) is dependent upon the values of defense variables, and is written as ϒ (W).

min ( ), Y f Y (4.1a) max min ( , ), Y X  f X Y (4.1b) ( )

min max min ( , , ).

W X YW f W X Y (4.1c)

The feasible region (ϒ (W)) is the same as Chapter I, except that it contains a single, additional side constraint limiting the time duration of any feasible path to a time budget (T), shown in Equation (4.1d).

( , ) . ij ijd i j A d D t Y T    

 

The side constraint rescinds the network structure of the problem, and the path variables (Y) must then be treated explicitly as binary variables. Consequently, the

constraint set for the inner operator model (ϒ (W)) does not allow for the direct substitution of continuous variables for the binary path decision variables (Y) as seen in Chapter 1. The formulation in (4.1c) is extremely difficult to solve.

In order to relieve the difficulty in (4.1c), we begin by isolating the operator model by fixing the values of the defense variables (W) and attack variables (X) with the use of the hat notation (^) in (4.1e).

ˆ ( )

ˆ ˆ

min ( , , ).

YW f W X Y (4.1e)

Ahuja et al. (1993, pp. 599–605) shows that Lagrangian relaxation of a side constraint can be used to regain the network structure of a CSP problem. The resulting heuristic algorithm can be converted to a competitive optimal algorithm as shown in Carlyle, Royset and Wood (2008).

We relax some of the constraints of the operator model (ϒ (W)) by incorporating them into a modified objective function with a Lagrange multiplier (μ) that penalizes infeasibility. We use the tilde notation (~) to signify the modification of the objective function (f) and operator model constraint set (ϒ (W)) in Equation (4.1f). Specifically, we use ( )Wˆ to represent the feasible region of a shortest path without the side constraint. We use fto represent the objective function with the penalized side constraint.

ˆ ˆ

( ) ( ) _{( , )}

ˆ ˆ ˆ ˆ ˆ ˆ

( , , ) min ( , , , ) min ( , , ) _{ij ijd} .

Y W Y W _{i j A d D} L W X  f W X Y  f W X Y  t Y T   _{ }      _   _ 

 

We can solve the Lagrangian multiplier problem by choosing the value of the multiplier that maximizes the Lagrangian multiplier function (L), as shown in (4.1g) (Ahuja et al., 1993, p. 608).

0

ˆ ˆ max ( , , ).L W X

The maximization of the Lagrange multiplier function (L) can be accomplished various ways. One way is to iteratively update the value of the Lagrange multiplier (μ) based on a subgradient calculation for a fixed defense (W) and attack (X). This is frequently coupled with a trust region method to force termination. See Ahuja et al. (1993, pp. 608–615) for a complete discussion. Since bounds of Lagrangian relaxation do not always converge, we employ a path enumeration technique of Carlyle, Royset and Wood (2008) to close the optimality gap.

We incorporate the Lagrange multiplier function maximization of (4.1g) into the tri-level model to see the effect of the relaxation on the DAD CSP problem in (4.1h).

( ) 0

min max max min ( , , , ) .

W X  YW f W X Y 

 

 

    (4.1h)

An alternative method to maximize the Lagrange multiplier function (L) is to focus on the bi-level attacker-defender (AD) model. For the AD version of CSP where defenses (W) are fixed, we take advantage of the side by side relationship of the two maximization operators in (4.1h). We can characterize the Lagrange multiplier (μ) as a decision variable, and combine the maximization operators. We define the bi-level AD CSP Lagrange variable problem in (4.1i) to use the Lagrange multiplier (μ) as a decision variable. Essentially, the attacker gets to choose the arcs to attack (X) and the penalties (μ) that the operator incurs for infeasibility of the side constraint.

ˆ , ( ) : 0 ˆ max min ( , , , ). X Y W f W X Y         (4.1i)

Since the relaxation of the side constraint allows us to treat the path choice variables (Y) as continuous variables, we can construct the dual formulation of the innermost minimization operator from (4.1i). The Lagrangian relaxation of bi-level AD CSP can be rewritten as a bi-level optimization model in (4.1j) by defining dual variables to represent the nodes (π) and arcs (α) of the network. We define (4.1j) as the Lagrange

X, , , : 0 ˆ max f W X( , , , , ).           (4.1j)

We can incorporate the Lagrange variable dual ILP problem into the tri-level model to see the effect of the relaxation on the DAD CSP problem in (4.1k).

X, , , : 0 min max ( , , , , ). W   f W X          (4.1k)

Expression (4.1k) is a relaxation of the original DAD CSP problem of (4.1c). Therefore, (4.1k) is not guaranteed to produce optimal solutions. Decomposition of (4.1k) over a series of iterations produces an AD sub problem in (4.1l) where the defense (W) is held constant. Decomposition produces a DAD master problem in (4.1m) where the attack (X), dual (α, π) and Lagrange (μ) variables are held constant.

, , :, : 0 ˆ max ( , , , , ), X  f W X         (4.1l) ˆ _{ˆ ˆ ˆ} min ( , , , , ). W f W X    (4.1m)

Decomposition produces valid lower bounds on the original DAD CSP problem instance because it incorporates Lagrangian relaxation. The decomposition in (4.1l) and (4.1m) produce heuristic solutions to DAD CSP instances more quickly than (4.1c) or (4.1h). We develop algorithms to incorporate the efficiencies within (4.1k) to produce optimal or provably near optimal solutions to DAD CSP instances more quickly than traditional nested decomposition techniques applied to (4.1c).

Proposition 4–1. DAD CSP is NP-Hard.

Proof of Proposition 4–1: When defense and attack budgets are restricted to zero units, DAD CSP reduces to the decision version of CSP, which is NP-Complete (Ahuja et al., 1993, p. 798). Therefore, DAD CSP is NP-Hard.

Note that if the defense budget is restricted to zero units and the side constraint time budget is not binding, then DAD CSP reduces to the network interdiction SP problem, the decision version of which is also NP-Complete (Wood, 1993). Evaluating a

feasible defense requires solving an NP-Complete AD problem. Finding the optimal defense adds an additional layer of complexity on top of an NP-Complete problem. It is unlikely that DAD CSP is even in NP.

Over the last few years, algorithms for solving instances of DAD SP have appeared in the literature. Alderson et al. (2011) gives a higher-level explanation for decomposition techniques for the tri-level optimization model for DAD SP. Alderson et al. (2014) provides a tutorial on the mathematical details for solving the tri-level DAD SP optimization model with decomposition procedures. To the best of our knowledge, the existing literature does not address the tri-level DAD CSP optimization problem, nor any solution procedures specifically developed for it.

The intent of this chapter is to investigate the application of Lagrangian relaxation to the DAD model for the CSP problem. We apply the nested decomposition methods found in Alderson et al. (2014 and 2015) to solve DAD CSP problem instances on small to medium sized test networks. We develop new heuristic methods to quickly find an estimated solution to DAD CSP instance using Lagrangian relaxation of the side constraint. We combine the heuristic methods with nested decomposition to obtain provably optimal solutions faster than using nested decomposition by itself. We apply our methods to a real-world problem to demonstrate scalability to large networks.