Parameterized Resiliency Problems via Integer Linear Programming

We introduce a framework in parameterized algorithms whose purpose is to solve resiliency versions of decision problems. In resiliency problems, the goal is to decide whether an instance remains positive after any (appropriately defined) perturbation has been applied to it. To tackle these kinds of problems, some of which might be of practical interest, we introduce a notion of resiliency for Integer Linear Programs (ILP). We prove that ILP Resiliency is fixed-parameter tractable (FPT) under a certain parameterization. To demonstrate the utility of our result, we prove that resiliency versions of several concrete problems are FPT under natural parameterizations. Our first result, for a problem which is of interest in access control, subsumes several FPT results and solves an open question from Crampton et al. (AAIM 2016). The second concerns the Closest String problem, for which we identify and solve two different resiliency problems, extending an FPT result of Gramm et al. (2003). We also consider problems in the fields of scheduling and social choice. We believe that many other problems can be tackled by our framework.