Parameterized Resiliency Problems via Integer Linear Programming

TL;DR

This paper introduces a framework for resiliency in decision problems, particularly ILP, demonstrating fixed-parameter tractability for various natural problems and extending existing results to new problem variants.

Contribution

It develops a general ILP resiliency framework using parametric linear programming and applies it to multiple problems, establishing their fixed-parameter tractability.

Findings

ILP resiliency is fixed-parameter tractable under certain parameters.

Resiliency versions of Disjoint Set Cover and Closest String are FPT.

Framework can be applied to scheduling and social choice problems.

Abstract

We introduce an extension of decision problems called resiliency 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) and show how to use a result of Eisenbrand and Shmonin (Math. Oper. Res., 2008) on Parametric Linear Programming to prove that ILP Resiliency is fixed-parameter tractable (FPT) under a certain parameterization. To demonstrate the utility of our result, we consider natural resiliency versions of several concrete problems, and prove that they are FPT under natural parameterizations. Our first results concern a four-variate problem which generalizes the Disjoint Set Cover problem and which is of interest in access control. We obtain a complete parameterized complexity classification for every possible combination of the parameters. Then, we introduce and study a resiliency version of the Closest String problem, for which we extend an FPT result of Gramm et al. (Algorithmica, 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.