This chapter extends network interdiction to directly include node interdic- tion. Current interdiction literature focuses primarily on arcs/edges. However, the
military often targets nodes. In addition, social networks revolve around people as nodes connected by relationships. Conceptually, it is often easier to think of
targeting/influencing people as opposed to nodes. Traditional network interdiction generally incorporates nodes by replacing each node with two artificial nodes and an
artificial link and then uses a links interdiction approach. However, this increases the size of the network (in fact, it would double the the number of nodes in the network
if every node is targetable), and in some cases, may not be intuitive to the user. To more transparently and directly represent nodes to target/protect, a formulation
that explicitly considers nodes is developed and discussed.
3.2 Background
Joint doctrine defines interdiction as follows:
Interdiction operations are actions to divert, disrupt, delay, or destroy an enemy’s surface capabilities before they can be used effectively against friendly forces, or to otherwise achieve objectives. [97, p. I-1]
Diversion is defined as making the enemy “consume resources or capabilities critical
to enemy operations in a way that is advantageous to friendly operations.” [97, p. I-2] Disruption involves “upsetting the flow of information, operational tempo,
effective interaction, or cohesion of the enemy force or those systems.” [97, p. I- 1] Delay involves “alter the ability of the enemy or adversary to project forces or
capabilities” in a timely manner. [97, p. I-3] Finally, destroy refers to “damage the structure, function, or condition of a target so that it can neither perform as
intended nor be restored to a usable condition, rendering it ineffective or useless.”
[97, p. I-4]
Since the term “interdiction” incorporates all these actions, some confusion may result if the terms are not used carefully. For example, if a node is destroyed,
then all flow through arcs to and from the destroyed node are disrupted. Technically, both the node and the arcs are “interdicted,” but using the specific terms such as
destroyed and/or disrupted clarifies how the interdiction has occurred (i.e. first-order versus second-order effects). These distinctions are rarely made in the operations
research literature, which makes interpretation of results difficult.
Interdiction doctrine also recognizes that interdiction may be potentially used for homeland security. “Interdiction also can be used to prevent an enemy from
achieving a variety of objectives affecting the US populace, economy, or national interests.” [97, p. I-1] In addition, doctrine also recognizes that information opera-
tions may play a unique role in interdiction stating, “interdiction may support, be supported by, or include aspects of information operations.” [97, p. I-1]
The nonlethal nature of many IO capabilities allows their use prior to and after hostilities, extending contact across time, thereby giving the friendly force greater opportunity to influence events and outcomes favorably. [97, p. II-13]
This chapter focuses on extensions to maximum flow interdiction to explic- itly allow nodal interdiction. Other network flow formulation could also be used.
For example, social network analysis relies heavily on shortest paths, so a shortest- path nodal interdiction is desired. Later chapters, Social Network Interdiction and
Multiple Layered Network Interdiction, demonstrate how nodal interdiction can be modified and incorporated to extend these formulations. However, for ease of illus-
tration, the primary focus of this chapter is maximum flow nodal interdiction.
Sections 2.2.3 and Section 2.5.2 discussed maximum flow and maximum flow arc-interdiction, respectively. The maximum flow problem can be written as
max xts s.t. X j xsj − X j xjs− xts = 0 X j xij − X j xji = 0, ∀ (i, j) ∈ A X j xtj− X j xjt+ xts = 0 0 ≤ xij ≤ uij, ∀ (i, j) ∈ A xij ≥ 0 (3.1)
where A is a set of directed arcs, xij represents flow across arc (i, j), xts is an artificial
arc from t to s, and uij is an upper bound on the flow across arc (i, j).
An attacker on this maximum flow network would seek to minimize the amount
of flow from s to t. If the attacker has sufficient resources, then he may determine a (node or mixed) cut-set (see Section 2.2.5) and prevent all flow from s to t. However,
if the attacker has limited resources, then he or she must determine which subset of elements to attack which would maximally limit the amount of flow. Wood has
shown that interdiction of this maximum flow can be expressed as the following bilevel program: [137, p. 5] min γ∈Γmaxx xts s.t. X j xsj − X j xjs− xts = 0 X j xij − X j xji = 0, ∀ i ∈ N-{s,t} (3.2) X j xtj − X j xjt + xts = 0 xij− uij(1 − γij) ≤ 0, ∀ (i, j) ∈ A xij ≥ 0, ∀(i, j) ∈ A ∪ {(t, s)} where Γ ≡ {γij|γij ∈ {0, 1}∀(i, j) ∈ A, P (i,j)∈A rijγij ≤ R}.
This formulation determines the set of arcs (identified by γij) whose disruptions
will maximally reduces the amount of resulting flow through the network. When γij = 1 then the 4th constraint of (3.2) forces the upper bound of flow through
that arc to 0; whereas when γij = 0 then flow (and upper bounds) on an arc are
unaffected.
To facilitate solution algorithms, Wood proves the model given in (3.2) can be
transformed into a single level integer program. He shows that for fixed γij, the inner
(follower’s) problem is a traditional maximum flow problem. Since maximum flow
problems are unimodular, the integrality requirements are non-restrictive, allowing the dual of this inner problem (with fixed γij) to be taken. Therefore, the inner prob-
lem can be replaced with its dual, and γij is “released.” With this transformation,
both objective functions are minimizations, but over differing sets of variables (the
leader’s primary variables, and the follower’s dual variables). The objective func- tions can, therefore, be combined resulting in the following mixed-integer program:
[137, p. 7] min X (i,j)∈A uijβij s.t. αi− αj+ βij + γij ≥ 0, ∀(i, j) ∈ A αt− αs ≥ 1 (3.3) X (i,j)∈A rijγij ≤ R αi ∈ {0, 1}, ∀i ∈ N βij, γij ∈ {0, 1}, ∀(i, j) ∈ A
where αi = 1 for i on the t side of the cut and αi = 0 for i on the s side of the cut;
γij is 1 if (i, j) is a forward arc across the cut which is to be disrupted; βij is 1 if
(i, j) is a forward arc across the cut, but is not to be disrupted; and all other βij and
As with formulation (3.2), the solution of Wood’s model in (3.3) indicates
which arcs (identified by γij = 1) should be denied to minimize the maximum flow
through the network.