Probability Distributions on Partially Ordered Sets and Network Interdiction Games

This article poses the following problem: Does there exist a probability distribution over subsets of a finite partially ordered set (poset), such that a set of constraints involving marginal probabilities of the poset's elements and maximal chains is satisfied? We present a combinatorial algorithm to positively resolve this question. The algorithm can be implemented in polynomial time in the special case where maximal chain probabilities are affine functions of their elements. This existence problem is relevant for the equilibrium characterization of a generic strategic interdiction game on a capacitated flow network. The game involves a routing entity that sends its flow through the network while facing path transportation costs, and an interdictor who simultaneously interdicts one or more edges while facing edge interdiction costs. Using our existence result on posets and strict complementary slackness in linear programming, we show that the Nash equilibria of this game can be fully described using primal and dual solutions of a minimum-cost circulation problem. Our analysis provides a new characterization of the critical components in the interdiction game. It also leads to a polynomial-time approach for equilibrium computation

Optimal Interdiction of Illegal Network Flow

Large scale smuggling of illegal goods is a longstanding problem, with $1.4b and thousands of agents assigned to protect the borders from such activity in the US-Mexico border alone. Illegal smuggling activities are usually blocked via inspection stations or ad-hoc checkpoints/roadblocks. Security resources are insufficient to man all stations at all times; furthermore, smugglers regularly conduct surveillance activities. This paper makes several contributions toward the challenging task of optimally interdicting an illegal network flow: i) A new Stackelberg game model for network flow interdiction; ii) A novel Column and Constraint Generation approach for computing the optimal defender strategy; iii) Complexity analysis of the column generation subproblem; iv) Compact convex nonlinear programs for solving the subproblems; v) Novel greedy and heuristic approaches for subproblems with good approximation guarantee. Experimental evaluation shows that our approach can obtain a robust enough solution outperforming the existing methods and heuristic baselines significantly and scale up to realistic-sized problems.NRF (Natl Research Foundation, S’pore)Published versio