Integer Programming and Incidence Treedepth

  • Eduard Eiben
  • Robert Ganian
  • Dušan Knop
  • Sebastian OrdyniakEmail author
  • Michał Pilipczuk
  • Marcin Wrochna
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11480)


Recently a strong connection has been shown between the tractability of integer programming (IP) with bounded coefficients on the one side and the structure of its constraint matrix on the other side. To that end, integer linear programming is fixed-parameter tractable with respect to the primal (or dual) treedepth of the Gaifman graph of its constraint matrix and the largest coefficient (in absolute value). Motivated by this, Koutecký, Levin, and Onn [ICALP 2018] asked whether it is possible to extend these result to a more broader class of integer linear programs. More formally, is integer linear programming fixed-parameter tractable with respect to the incidence treedepth of its constraint matrix and the largest coefficient (in absolute value)?

We answer this question in negative. We prove that deciding the feasibility of a system in the standard form, \({A\mathbf {x}= \mathbf {b}}, {\mathbf {l} \le \mathbf {x}\le \mathbf {u}}\), is NP-hard even when the absolute value of any coefficient in A is 1 and the incidence treedepth of A is 5. Consequently, it is not possible to decide feasibility in polynomial time even if both the assumed parameters are constant, unless \(\mathsf {P}=\mathsf {NP}\).


Integer programming Incidence treedepth Gaifman graph Computational complexity 


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Eduard Eiben
    • 1
  • Robert Ganian
    • 2
  • Dušan Knop
    • 3
    • 4
  • Sebastian Ordyniak
    • 5
    Email author
  • Michał Pilipczuk
    • 6
  • Marcin Wrochna
    • 6
    • 7
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.Algorithms and Complexity GroupTechnische Universität WienViennaAustria
  3. 3.Algorithmics and Computational Complexity, Faculty IVTU BerlinBerlinGermany
  4. 4.Department of Theoretical Computer Science, Faculty of Information TechnologyCzech Technical University in PraguePragueCzech Republic
  5. 5.Algorithms Group, Department of Computer ScienceUniversity of SheffieldSheffieldUK
  6. 6.Institute of InformaticsUniversity of WarsawWarsawPoland
  7. 7.University of OxfordOxfordUK

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