On Integer Programs That Look Like Paths

2025-10-25

AI Summary

This paper studies the feasibility problem of integer programs (IPs) whose constraint matrices exhibit a *path structure*—i.e., each column has nonzero entries in at most two consecutive rows. Despite restricting all coefficient absolute values to at most 8, we prove the problem is NP-hard—the first lower bound establishing computational intractability for path-structured IPs with small coefficients. Our proof employs a carefully engineered polynomial-time reduction from 3-SAT, constructing constraint matrices that strictly satisfy the path structure while embedding clause-variable dependencies via tailored coefficient patterns. Combining combinatorial structural analysis with complexity-theoretic reasoning, we refute the intuitive conjecture that path-structured IPs should be as tractable as star-structured ones. This result delineates a new hardness frontier for structured integer programming: even under strong structural constraints and bounded small coefficients, polynomial-time solvability remains impossible. The finding provides foundational insights for structural optimization and algorithm design in discrete optimization.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Structured Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Web Mining and Content Analysis: Bridging structured and unstructured data
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Solving integer programs of the form $min {mathbf{x} mid Amathbf{x} = mathbf{b}, mathbf{l} leq mathbf{x} leq mathbf{u}, mathbf{x} in mathbb{Z}^n }$ 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.

Problem

• Solving integer programs with path-like constraint matrix structure
• Proving NP-hardness for path-structured integer programs
• Analyzing computational complexity when coefficients are bounded

Innovation

• Studies integer programs with path-like constraint matrices
• Proves NP-hardness for coefficients bounded by eight
• Uses reduction from 3-SAT for hardness proof