Integer Programming and Incidence Treedepth

TL;DR

This paper investigates the complexity of integer programming related to the structure of its constraint matrix, proving NP-hardness for certain parameters and identifying specific cases where the problem remains tractable.

Contribution

It extends the understanding of fixed-parameter tractability in integer programming by showing NP-hardness with incidence treedepth constraints and identifying tractable special cases.

Findings

Deciding feasibility is NP-hard with incidence treedepth 5 and coefficients of absolute value 1.

Polynomial-time feasibility decision is unlikely unless P=NP.

Certain restricted settings, like bounded treedepth with additional constraints, are tractable.

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, Kouteck\'y, 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. In particular, 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 $\mathsf{NP}$-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 $\mathsf{P}=\mathsf{NP}$. Moreover, we complement this intractability result by showing tractability for natural and only slightly more restrictive settings, namely: (1) treedepth with an additional bound on either the maximum arity of constraints or the maximum number of occurrences of variables and (2) the vertex cover number.