Integer Programming and Incidence Treedepth

Abstract

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.

Appendix

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 $A-v$.   $\square$