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On Integer Programs That Look Like Paths

Marcin Briański, Alexandra Lassota, Kristýna Pekárková, Michał Pilipczuk, Janina Reuter

TL;DR

The paper investigates integer programs with path-like constraint matrices, where each nonzero entry of $A$ lies in at most two consecutive constraints. It establishes NP-hardness for feasibility even when $\|A\|_\infty \le 8$ via a 3-SAT–driven layered reduction, and provides an ETH-based lower bound in that regime. By contrast, it identifies a tractable frontier: hollow staircase matrices with $A \in \{-1,0,1\}$ and $\mathbf{x} \ge 0$ allow polynomial-time minimization of a linear objective, supported by a universal $\ell_\infty$ bound of $2$ on Graver basis elements, enabling augmentation. The work also raises open questions about tightening the hardness constant below $8$ and the complexity of the unbounded-variable version, highlighting the nuanced boundary between tractability and hardness for path-like IPs.

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.

On Integer Programs That Look Like Paths

TL;DR

The paper investigates integer programs with path-like constraint matrices, where each nonzero entry of lies in at most two consecutive constraints. It establishes NP-hardness for feasibility even when via a 3-SAT–driven layered reduction, and provides an ETH-based lower bound in that regime. By contrast, it identifies a tractable frontier: hollow staircase matrices with and allow polynomial-time minimization of a linear objective, supported by a universal bound of on Graver basis elements, enabling augmentation. The work also raises open questions about tightening the hardness constant below and the complexity of the unbounded-variable version, highlighting the nuanced boundary between tractability and hardness for path-like IPs.

Abstract

Solving integer programs of the form is, in general, -hard. Hence, great effort has been put into identifying subclasses of integer programs that are solvable in polynomial or 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 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 are bounded by 8, deciding the feasibility of such integer programs is -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.
Paper Structure (3 sections, 3 theorems, 12 equations, 1 figure, 1 table)

This paper contains 3 sections, 3 theorems, 12 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Hardness
  3. Summary and Open Problems

Key Result

Theorem 1

The problem of deciding whether a given integer program $\{\mathbf{x}\,\mid\,A\mathbf{x}=\mathbf{b}$, $\mathbf{l}\leqslant \mathbf{x}\leqslant \mathbf{u}\}$ has a solution is $\mathsf{NP}$- hard even when $A$ is a path-like matrix with all entries of absolute value at most $8$. Here, we assume that

Figures (1)

  • Figure 1: A schematic depiction of a path-like constraint matrix $A$.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • proof
  • Claim 3
  • proof
  • Corollary 4
  • proof