Resolving Infeasibility of Linear Systems: A Parameterized Approach

Authors Alexander Göke, Lydia Mirabel Mendoza Cadena, Matthias Mnich

Thumbnail PDF


  • Filesize: 0.63 MB
  • 15 pages

Document Identifiers

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


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)


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
  • Infeasible subsystems
  • linear programming
  • fixed-parameter algorithms


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Edoardo Amaldi and Viggo Kann. On the approximability of minimizing nonzero variables or unsatisfied relations in linear systems. Theoret. Comput. Sci., 209(1-2):237-260, 1998. Google Scholar
  2. Sanjeev Arora, László Babai, Jacques Stern, and Z. Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. System Sci., 54(2, part 2):317-331, 1997. Google Scholar
  3. Jørgen Bang-Jensen and Gregory Z. Gutin. Digraphs: theory, algorithms and applications. Springer Science & Business Media, 2008. Google Scholar
  4. Piotr Berman and Marek Karpinski. Approximating Minimum Unsatisfiability of Linear Equations. In Proc. SODA 2002, pages 514-516, 2002. Google Scholar
  5. Hans L. Bodlaender, Fedor V. Fomin, and Saket Saurabh. Open problems from 2010 Workshop on Kernels, 2010. URL:
  6. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization Lower Bounds by Cross-Composition. SIAM J. Discrete Math., 28(1):277-305, 2014. Google Scholar
  7. Hans L. Bodlaender, Stéphan Thomassé, and Anders Yeo. Kernel bounds for disjoint cycles and disjoint paths. Theoret. Comput. Sci., 412(35):4570-4578, 2011. Google Scholar
  8. Marthe Bonamy, Łukasz Kowalik, Jesper Nederlof, Michał Pilipczuk, Arkadiusz Socała, and Marcin Wrochna. On Directed Feedback Vertex Set Parameterized by Treewidth. In Proc. WG 2018, pages 65-78, 2018. Google Scholar
  9. Nilotpal Chakravarti. Some results concerning post-infeasibility analysis. European J. Oper. Res., 73(1):139-143, 1994. Google Scholar
  10. Jianer Chen, Yang Liu, Songjian Lu, Barry O'Sullivan, and Igor Razgon. A fixed-parameter algorithm for the directed feedback vertex set problem. J. ACM, 55(5):Art. 21, 19, 2008. Google Scholar
  11. John W. Chinneck. An effective polynomial-time heuristic for the minimum-cardinality IIS set-covering problem. Ann. Math. Artif. Intell., 17(1):127-144, 1996. Google Scholar
  12. John W. Chinneck. Finding a Useful Subset of Constraints for Analysis in an Infeasible Linear Program. INFORMS J. Comput., 9(2):164-174, 1997. Google Scholar
  13. John W. Chinneck. Feasibility and infeasibility in optimization: algorithms and computational methods, volume 118 of International Series in Operations Research & Management Science. Springer, New York, 2008. Google Scholar
  14. Rajesh Chitnis, Marek Cygan, Mohammataghi Hajiaghayi, and Dániel Marx. Directed subset feedback vertex set is fixed-parameter tractable. ACM Trans. Algorithms, 11(4):Art. 28, 28, 2015. Google Scholar
  15. Gianni Codato and Matteo Fischetti. Combinatorial Benders' cuts for mixed-integer linear programming. Oper. Res., 54(4):756-766, 2006. Google Scholar
  16. Robert Crowston, Gregory Gutin, Mark Jones, and Anders Yeo. Parameterized complexity of satisfying almost all linear equations over 𝔽₂. Theory Comput. Syst., 52(4):719-728, 2013. Google Scholar
  17. Marek Cygan, Fedor Fomin, Bart M.P. Jansen, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Open Problems collected for the 2014 School on Parameterized Complexity in Bȩdlewo, Poland, 2014. URL:
  18. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized algorithms. Springer, 2015. Google Scholar
  19. Marek Cygan, Łukasz Kowalik, and Marcin Pilipczuk. Open problems from 2013 Workshop on Kernels, 2013. URL:
  20. Till Fluschnik, Danny Hermelin, André Nichterlein, and Rolf Niedermeier. Fractals for Kernelization Lower Bounds, With an Application to Length-Bounded Cut Problems. In Proc. ICALP 2016, volume 55 of Leibniz Int. Proc. Informatics, pages 25:1-25:14, 2016. Google Scholar
  21. Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, MichaŁPilipczuk, and Marcin Wrochna. Fully Polynomial-Time Parameterized Computations for Graphs and Matrices of Low Treewidth. ACM Trans. Algorithms, 14(3):34:1-34:45, 2018. Google Scholar
  22. Panos Giannopoulos, Christian Knauer, and Günter Rote. The parameterized complexity of some geometric problems in unbounded dimension. In Proc. IWPEC 2009, volume 5917 of Lecture Notes Comput. Sci., pages 198-209. 2009. Google Scholar
  23. Petr A. Golovach and Dimitrios M. Thilikos. Paths of bounded length and their cuts: Parameterized complexity and algorithms. Discrete Optim., 8(1):72-86, 2011. Google Scholar
  24. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric algorithms and combinatorial optimization, volume 2 of Algorithms and Combinatorics. Springer-Verlag, Berlin, second edition, 1993. Google Scholar
  25. Sylvain Guillemot. FPT algorithms for path-transversal and cycle-transversal problems. Discrete Optim., 8(1):61-71, 2011. Google Scholar
  26. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of computer computations (Proc. Sympos., IBM, Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), pages 85-103. Plenum, New York, 1972. Google Scholar
  27. Stefan Kratsch and Magnus Wahlström. Two edge modification problems without polynomial kernels. Discrete Optim., 10(3):193-199, 2013. Google Scholar
  28. Neele Leithäuser, Sven O. Krumke, and Maximilian Merkert. Approximating infeasible 2VPI-systems. In Proc. WG 2012, volume 7551 of Lecture Notes Comput. Sci., pages 225-236. 2012. Google Scholar
  29. Daniel Lokshtanov, M. S. Ramanuajn, and Saket Saurabh. When Recursion is Better Than Iteration: A Linear-time Algorithm for Acyclicity with Few Error Vertices. In Proc. SODA 2018, pages 1916-1933, 2018. Google Scholar
  30. Matthias Mnich and Erik Jan van Leeuwen. Polynomial kernels for deletion to classes of acyclic digraphs. Discrete Optim., 25:48-76, 2017. Google Scholar
  31. Marc E. Pfetsch. Branch-and-cut for the maximum feasible subsystem problem. SIAM J. Optim., 19(1):21-38, 2008. Google Scholar
  32. G. Ramalingam, J. Song, L. Joskowicz, and R. E. Miller. Solving Systems of Difference Constraints Incrementally. Algorithmica, 23(3):261-275, 1999. Google Scholar
  33. Jayaram K. Sankaran. A note on resolving infeasibility in linear programs by constraint relaxation. Oper. Res. Lett., 13(1):19-20, 1993. Google Scholar
  34. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer Science & Business Media, 2003. Google Scholar
  35. K. Subramani and Piotr Wojciechowski. A Combinatorial Certifying Algorithm for Linear Feasibility in UTVPI Constraints. Algorithmica, 78(1):166-208, 2017. Google Scholar
  36. Meirav Zehavi. Mixing color coding-related techniques. In Proc. ESA 2015, volume 9294 of Lecture Notes Comput. Sci., pages 1037-1049. 2015. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail