On the Complexity of Delaying an Adversary’s Project

2015

A “project manager ” wishes to complete a project (e.g., a weapons-development program) as quickly as possible. Using a limited interdic-tion budget, an “interdictor ” wishes to delay the project’s overall com-pletion time by interdicting and thereby delaying some of the project’s component tasks. We explore a variety of PERT-based interdiction models for such problems and show that the resulting problem com-plexities run the gamut: polynomially solvable, weakly NP-complete, strongly NP-complete or NP-hard. We suggest methods for solving the problems that are easier than worst-case complexity implies.

European Journal of Operational Research, 2014

We consider project scheduling where the project manager's objective is to minimize the time from when an adversary discovers the project until the completion of the project. We analyze the complexity of the problem identifying both polynomially solvable and NP-hard versions of the problem. The complexity of the problem is seen to be dependent on the nature of renewable resource constraints, precedence constraints, and the ability to crash activities in the project.

Operations Research, 2009

A "proliferator" seeks to complete a first small batch of fission weapons as quickly as possible, while an "interdictor" wishes to delay that completion for as long as possible. We develop and solve a max-min model that identifies resource-limited interdiction actions that maximally delay completion time of the proliferator's weapons project, given that the proliferator will observe any such actions and adjust his plans to minimize that time. The model incorporates a detailed project-management (CPM) submodel, and standard optimization software solves the model in a few minutes on a personal computer. We exploit off-the-shelf project-management software to manage a database, control the optimization, and display results. Using a range of levels for interdiction effort, we analyze a published case study that models three, alternate, uranium-enrichment technologies. The task of "cascade loading" appears in all technologies and turns out to be an inherent fragility for the proliferator at all levels of interdiction effort. Such insights enable policymakers to quantify the effects of interdiction options at their disposal, be they diplomatic, economic, or military.

RAIRO - Operations Research

There are two opponents in a classic network interdiction problem, network owner/defender and interdictor/attacker. Each side has enough information about the other’s possible courses of action. While the network user wishes to run the network in an optimal way, the attacker with the limited resources tries to prevent the optimal operation of the network by interdicting the arcs/nodes of the network. In this study, we investigate project management in a competitive environment using a network interdiction approach. We assume that the project owner/manager strives to minimize the completion time of a Critical Path Method (CPM) based project while an opponent attempts to maximize the minimum completion time by inflicting some delays on project activities with available interdiction resources. Considering both discrete and continuous delay times, we develop two bi-level mixed-integer programming models for the interdictor. Using duality, we then convert the bi-level models to standard ...

Operations Research Letters, 2016

Network interdiction can be viewed as a game between two players, an interdictor and a flow player. The flow player wishes to send as much material as possible through a network, while the interdictor attempts to minimize the amount of transported material by removing a certain number of arcs, say Γ arcs. We introduce the randomized network interdiction problem that allows the interdictor to use randomness to select arcs to be removed. We model the problem in two different ways: arc-based and pathbased formulations, depending on whether flows are defined on arcs or paths, respectively. We present insights into the modeling power, complexity, and approximability of both formulations. In particular, we prove that Z NI /Z RNI ≤ Γ + 1, Z NI /Z Path RNI ≤ Γ + 1, Z RNI /Z Path RNI ≤ Γ, where Z NI , Z RNI , and Z Path RNI are the optimal values of the network interdiction problem and its randomized versions in arc-based and path-based formulations, respectively. We also show that these bounds are tight. We show that it is NP-hard to compute the values Z RNI and Z Path RNI for a general Γ, but they are computable in polynomial time when Γ = 1. Further, we provide a (Γ + 1)-approximation for Z NI , a Γ-approximation for Z RNI , and a 1 + ⌊Γ/2⌋ • ⌈Γ/2⌉/(Γ + 1)-approximation for Z Path RNI .

2013

Operations Research Letters, 2007

An important topic in PERT networks is how to allocate the total expedition (or delay) for situations in which the project is not executed as planned. In order to do that we define a TU project game that satisfies some desirable properties from the management project and game theory point of view.

We address a PERT/CPM scheduling problem with AND/OR constraints and examine its relations with extremal problems in grammars and hypergraphs. We demonstrate that two scheduling algorithms developed independently by and have different worst-case complexity and, in a sense, are incomparable.

Lecture Notes in Computer Science, 2013

Interdiction problems are leader-follower games in which the leader is allowed to delete a certain number of edges from the graph in order to maximally impede the follower, who is trying to solve an optimization problem on the impeded graph. We introduce approximation algorithms and strong NP-completeness results for interdiction problems on planar graphs. We give a multiplicative (1 + ǫ)-approximation for the maximum matching interdiction problem on weighted planar graphs. The algorithm runs in pseudo-polynomial time for each fixed ǫ > 0. We also show that weighted maximum matching interdiction, budget-constrained flow improvement, directed shortest path interdiction, and minimum perfect matching interdiction are strongly NPcomplete on planar graphs. To our knowledge, our budget-constrained flow improvement result is the first planar NP-completeness proof that uses a one-vertex crossing gadget.

2005

We consider two network interdiction problems: one where a network user tries to traverse a network from a starting vertex s to a target vertex t along the shortest path while an interdictor tries to eliminate all short s-t paths by destroying as few vertices (arcs) as possible, and one where the network user, as before, tries to traverse the network from s to t along the shortest path while the interdictor tries to destroy a fixed number of vertices (arcs) so as to cause the biggest increase in the shortest s-t path. The latter problem is known as the Most Vital Vertices (Arcs) Problem. In this paper we provide inapproximability bounds for several variants of these problems.