Parameterized Complexity of Edge Interdiction Problems

TL;DR

This paper investigates the parameterized complexity of edge interdiction problems in graphs, analyzing their computational difficulty and fixed-parameter tractability for problems like minimum spanning tree, maximum matching, and shortest paths.

Contribution

It establishes new fixed-parameter tractability and hardness results for edge interdiction problems and explores their relation to partial cover problems on bipartite graphs.

Findings

Identifies fixed-parameter tractability for certain interdiction problems.

Proves W[1]-hardness for other interdiction problems.

Shows connections between interdiction and partial cover problems.

Abstract

We study the parameterized complexity of interdiction problems in graphs. For an optimization problem on graphs, one can formulate an interdiction problem as a game consisting of two players, namely, an interdictor and an evader, who compete on an objective with opposing interests. In edge interdiction problems, every edge of the input graph has an interdiction cost associated with it and the interdictor interdicts the graph by modifying the edges in the graph, and the number of such modifications is constrained by the interdictor's budget. The evader then solves the given optimization problem on the modified graph. The action of the interdictor must impede the evader as much as possible. We focus on edge interdiction problems related to minimum spanning tree, maximum matching and shortest paths. These problems arise in different real world scenarios. We derive several fixed-parameter tractability and W[1]-hardness results for these interdiction problems with respect to various parameters. Next, we show close relation between interdiction problems and partial cover problems on bipartite graphs where the goal is not to cover all elements but to minimize/maximize the number of covered elements with specific number of sets. Hereby, we investigate the parameterized complexity of several partial cover problems on bipartite graphs.