Continuous Length-Bounded Paths Interdiction

Network vulnerability assessment, in which a communication between nodes is functional if their distance under a given metric is lower than a pre-defined threshold, has received significant attention recently. However, those works only focused on discrete domain while many practical applications require us to investigate in the continuous domain. Motivated by this observation, we study a Length-bounded Paths Interdiction in Continuous Domain (cLPI) problem: given a network \(G=(V,E)\) , in which each edge \(e \in E\) is associated with a function \(f_e(x)\) in continuous domain, and a set of target pairs of nodes, find a distribution \(\textbf{x}: E \rightarrow \mathbb {R}^\ge \) with minimum \(\sum _{e \in E} \textbf{x}(e)\) that ensures any path p, connecting a target pair, satisfies \( \sum _{e \in p} f_e(\textbf{x}(e)) \ge T\) . We first propose a general framework to solve cLPI by designing two oracles, namely Threshold Blocking (TB) oracle and Critical Path Listing (CPL) oracle, which communicate back and forth to construct a feasible solution with theoretical performance guarantees. Based on this framework, we propose a bicriteria approximation algorithm to cLPI. This bicriteria guarantee allows us to control the solutions’s trade-off between the running time and the performance accuracy.