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Integer Programming and Incidence Treedepth

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Integer Programming and Combinatorial Optimization (IPCO 2019)


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 \(\mathsf {P}=\mathsf {NP}\).

Eduard Eiben was supported by Pareto-Optimal Parameterized Algorithms (ERC Starting Grant 715744). This work is a part of projects CUTACOMBS, PowAlgDO (M. Wrochna) and TOTAL (M. Pilipczuk) that have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements No. 714704, No. 714532, and No. 677651). Dušan Knop is supported by DFG, project “MaMu”, NI 369/19. Marcin Wrochna is supported by Foundation for Polish Science (FNP) via the START stipend. Robert Ganian acknowledges support from the FWF Austrian Science Fund (Project P31336: NFPC) and is also affiliated with FI MU, Brno, Czech Republic.

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  1. 1.

    That is, an algorithm running in time \(f(r,s,\Vert E\Vert _\infty ) \cdot n^{O(1)}L\).


  1. Altmanová, K., Knop, D., Koutecký, M.: Evaluating and tuning n-fold integer programming. In: D’Angelo, G. (ed.) 17th International Symposium on Experimental Algorithms, SEA 2018, L’Aquila, Italy, 27–29 June 2018. LIPIcs, vol. 103, pp. 10:1–10:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018).

  2. Chatzigiannakis, I., Kaklamanis, C., Marx, D., Sannella, D. (eds.): 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, Prague, Czech Republic, 9–13 July 2018. LIPIcs, vol. 107. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018).

  3. Chen, L., Xu, L., Shi, W.: On the graver basis of block-structured integer programming. CoRR abs/1805.03741 (2018).

  4. De Loera, J.A., Hemmecke, R., Köppe, M.: Algebraic and Geometric Ideas in the Theory of Discrete Optimization. MOS-SIAM Series on Optimization, vol. 14. SIAM (2013).

  5. Dechter, R.: Chapter 7 - tractable structures for constraint satisfaction problems. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming, Foundations of Artificial Intelligence, vol. 2, pp. 209–244. Elsevier (2006).

    Chapter  Google Scholar 

  6. Eiben, E., Ganian, R., Knop, D., Ordyniak, S.: Unary integer linear programming with structural restrictions. In: Lang, J. (ed.) Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, IJCAI 2018, Stockholm, Sweden, 13–19 July 2018, pp. 1284–1290. (2018).

  7. Eisenbrand, F., Hunkenschröder, C., Klein, K.: Faster algorithms for integer programs with block structure. In: Chatzigiannakis et al. [2], pp. 49:1–49:13.

  8. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)

    Google Scholar 

  9. Hemmecke, R., Köppe, M., Weismantel, R.: A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs. In: Eisenbrand, F., Shepherd, F.B. (eds.) IPCO 2010. LNCS, vol. 6080, pp. 219–229. Springer, Heidelberg (2010).

    Chapter  MATH  Google Scholar 

  10. Hemmecke, R., Onn, S., Romanchuk, L.: N-fold integer programming in cubic time. Math. Program. 137(1–2), 325–341 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  11. Hemmecke, R., Schultz, R.: Decomposition of test sets in stochastic integer programming. Math. Program. 94(2), 323–341 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  12. Jansen, K., Lassota, A., Rohwedder, L.: Near-linear time algorithm for n-fold ILPs via color coding. CoRR abs/1811.00950 (2018)

    Google Scholar 

  13. Klein, K.: About the complexity of two-stage stochastic IPs. CoRR abs/1901.01135 (2019).

  14. Koutecký, M., Levin, A., Onn, S.: A parameterized strongly polynomial algorithm for block structured integer programs. In: Chatzigiannakis et al. [2], pp. 85:1–85:14.

  15. Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  16. Nešetřil, J., Ossona de Mendez, P.: Sparsity - Graphs, Structures, and Algorithms. AC, vol. 28. Springer, Heidelberg (2012).

    Book  MATH  Google Scholar 

  17. Onn, S.: Nonlinear Discrete Optimization: An Algorithmic Theory (Zurich Lectures in Advanced Mathematics). European Mathematical Society Publishing House (2010)

    Google Scholar 

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Proof of Lemma

4 . We prove only the second inequality, as the first one is symmetric. The proof is by induction with respect to the total number of rows and columns of the matrix A. The base of the induction, when A has one row and one column, is trivial, so we proceed to the induction step.

Observe that \(G_I(A)\) is disconnected if and only if \(G_D(A)\) is disconnected if and only if A is a block-decomposable matrix. Moreover, the incidence treedepth of A is the maximum incidence treedepth among the blocks of A, and the same also holds for the dual treedepth. Hence, in this case we may apply the induction hypothesis to every block of A and combine the results in a straightforward manner.

Assume then that \(G_I(A)\) is connected. Then

$${{\,\mathrm{{\text {td}}}\,}}(G_I(A))=1+\min _{v\in V(G_I(A))} {{\,\mathrm{{\text {td}}}\,}}(G_I(A)-v).$$

Let v be the vertex for which the minimum on the right hand side is attained. We consider two cases: either v is a row of A or a column of A.

Suppose first that v is a row of A. Then we have

$$\begin{aligned} {{\,\mathrm{{\text {td}}}\,}}(G_D(A))\le & {} 1+{{\,\mathrm{{\text {td}}}\,}}(G_D(A)-v)\\\le & {} 1+{{\,\mathrm{{\text {maxdeg}}}\,}}_V(A)\cdot {{\,\mathrm{{\text {td}}}\,}}(G_I(A)-v)\\= & {} 1+{{\,\mathrm{{\text {maxdeg}}}\,}}_V(A)\cdot ({{\,\mathrm{{\text {td}}}\,}}(G_I(A))-1)\\\le & {} {{\,\mathrm{{\text {maxdeg}}}\,}}_V(A)\cdot {{\,\mathrm{{\text {td}}}\,}}(G_I(A)) \end{aligned}$$

as required, where the second inequality follows from applying the induction assumption to A with the row v removed.

Finally, suppose that v is a column of A. Let X be the set of rows of A that contain non-zero entries in column v; then \(|X|\le {{\,\mathrm{{\text {maxdeg}}}\,}}_V(A)\) and X is non-empty, because \(G_I(A)\) is connected. If we denote by \(A-v\) the matrix obtained from A by removing column v, then we have

$$\begin{aligned} {{\,\mathrm{{\text {td}}}\,}}(G_D(A))\le & {} |X|+{{\,\mathrm{{\text {td}}}\,}}(G_D(A)-X) \\\le & {} {{\,\mathrm{{\text {maxdeg}}}\,}}_V(A)+{{\,\mathrm{{\text {td}}}\,}}(G_D(A-v))\\\le & {} {{\,\mathrm{{\text {maxdeg}}}\,}}_V(A)+{{\,\mathrm{{\text {maxdeg}}}\,}}_V(A)\cdot {{\,\mathrm{{\text {td}}}\,}}(G_I(A-v))\\\le & {} {{\,\mathrm{{\text {maxdeg}}}\,}}_V(A)\cdot {{\,\mathrm{{\text {td}}}\,}}(G_I(A)), \end{aligned}$$

as required. Here, in the second inequality we used the fact that \(G_D(A)-X\) is a subgraph of \(G_D(A-v)\), while in the third inequality we used the induction assumption for the matrix \(A-v\).   \(\square \)

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Eiben, E., Ganian, R., Knop, D., Ordyniak, S., Pilipczuk, M., Wrochna, M. (2019). Integer Programming and Incidence Treedepth. In: Lodi, A., Nagarajan, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2019. Lecture Notes in Computer Science(), vol 11480. Springer, Cham.

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