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Resolving Infeasibility of Linear Systems: A Parameterized Approach

Authors Alexander Göke, Lydia Mirabel Mendoza Cadena, Matthias Mnich

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Author Details

Alexander Göke
  • Universität Bonn, Bonn, Germany
  • Technische Universität Hamburg, Hamburg, Germany
Lydia Mirabel Mendoza Cadena
  • Eötvös Loránd University, Budapest, Hungary
Matthias Mnich
  • Universität Bonn, Bonn, Germany
  • Technische Universität Hamburg, Hamburg, Germany

Funding

  • Göke, Alexander: Supported by DAAD with funds of the Bundes- ministerium für Bildung und Forschung (BMBF) and by DFG project MN 59/1-1.
  • Mnich, Matthias: Supported by DAAD with funds of the Bundesministerium für Bildung und Forschung (BMBF) and by DFG project MN 59/4-1.

Acknowledgements

We thank an anonymous reviewer of an earlier version for a suggestion on run time improvements.

Cite AsGet BibTex

Alexander Göke, Lydia Mirabel Mendoza Cadena, and Matthias Mnich. Resolving Infeasibility of Linear Systems: A Parameterized Approach. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.IPEC.2019.17

Abstract

Deciding feasibility of large systems of linear equations and inequalities is one of the most fundamental algorithmic tasks. However, due to inaccuracies of the data or modeling errors, in practical applications one often faces linear systems that are infeasible. Extensive theoretical and practical methods have been proposed for post-infeasibility analysis of linear systems. This generally amounts to detecting a feasibility blocker of small size k, which is a set of equations and inequalities whose removal or perturbation from the large system of size m yields a feasible system. This motivates a parameterized approach towards post-infeasibility analysis, where we aim to find a feasibility blocker of size at most k in fixed-parameter time f(k)* m^{O(1)}. On the one hand, we establish parameterized intractability (W[1]-hardness) results even in very restricted settings. On the other hand, we develop fixed-parameter algorithms parameterized by the number of perturbed inequalities and the number of positive/negative right-hand sides. Our algorithms capture the case of Directed Feedback Arc Set, a fundamental parameterized problem whose fixed-parameter tractability was shown by Chen et al. (STOC 2008).

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Infeasible subsystems
  • linear programming
  • fixed-parameter algorithms

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