On Integer Programs That Look Like Paths

TL;DR

This paper investigates the computational complexity of a specific class of integer programs with path-like constraint matrices, revealing that even with bounded coefficients, the feasibility problem remains NP-hard, contrasting with simpler star-like structures.

Contribution

It proves NP-hardness of feasibility for path-structured integer programs with bounded coefficients, highlighting a complexity boundary in structured integer programming.

Findings

Feasibility is NP-hard even with coefficients bounded by 8.

Path-like structure does not guarantee polynomial-time solvability.

Contrasts with polynomial algorithms for star-like and column-bounded structures.

Abstract

Solving integer programs of the form $\min \{\mathbf{x} \mid A\mathbf{x} = \mathbf{b}, \mathbf{l} \leq \mathbf{x} \leq \mathbf{u}, \mathbf{x} \in \mathbb{Z}^n \}$ is, in general, $\mathsf{NP}$-hard. Hence, great effort has been put into identifying subclasses of integer programs that are solvable in polynomial or $\mathsf{FPT}$ time. A common scheme for many of these integer programs is a star-like structure of the constraint matrix. The arguably simplest form that is not a star is a path. We study integer programs where the constraint matrix $A$ has such a path-like structure: every non-zero coefficient appears in at most two consecutive constraints. We prove that even if all coefficients of $A$ are bounded by 8, deciding the feasibility of such integer programs is $\mathsf{NP}$-hard via a reduction from 3-SAT. Given the existence of efficient algorithms for integer programs with star-like structures and a closely related pattern where the sum of absolute values is column-wise bounded by 2 (hence, there are at most two non-zero entries per column of size at most 2), this hardness result is surprising.