Diagonal Frobenius Number via Gomory's Relaxation and Discrepancy

2025-09-06

AI Summary

This paper studies the diagonal Frobenius number $ F_{ ext{diag}}(A) $ of an integer matrix $ A in mathbb{Z}^{m imes n} $, defined as the smallest integer $ t $ such that for every $ b in operatorname{cone}_mathbb{Z}(A) $, if $ b = Ax $ admits a nonnegative real solution $ x geq t mathbf{1} $, then $ b = Az $ has a nonnegative integer solution $ z in mathbb{Z}_{geq 0}^n $. The authors introduce the generalized slack Frobenius number $ F_{ ext{slack}}(A) $ by unifying Gomory’s corner polyhedron relaxation with discrepancy theory. They establish a tight bound $ F_{ ext{diag}}(A) = Delta + O(log k) $, where $ Delta $ is the maximum absolute value of all $ k imes k $ subdeterminants of $ A $ and $ k = operatorname{rank}(A) $. Under preprocessing or when a basis is known, this bound is optimal. Moreover, they present a polynomial-time algorithm achieving the $ O(Delta log k) $ bound.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

For a matrix $A in Z^{k imes n}$ of rank $k$, the diagonal Frobenius number $F_{ ext{diag}}(A)$ is defined as the minimum $t in Z_{geq 1}$, such that, for any $b in ext{span}_{Z}(A)$, the condition egin{equation*} exists x in R_{geq 0}^n,, x geq t cdot 1 colon quad b = A x end{equation*} implies that egin{equation*} exists z in Z_{geq 0}^n colonquad b = A z. end{equation*} In this work, we show that egin{equation*} F_{ ext{diag}}(A) = Δ+ O(log k), end{equation*} where $Δ$ denotes the maximum absolute value of $k imes k$ sub-determinants of $A$. From the computational complexity perspective, we show that the integer vector $z$ can be found by a polynomial-time algorithm for some weaker values of $t$ in the described condition. For example, we can choose $t = O( Δcdot log k)$ or $t = Δ+ O(sqrt{k} cdot log k)$. Additionally, in the assumption that a $2^k$-time preprocessing is allowed or a base $J$ with $|{det A_{J}}| = Δ$ is given, we can choose $t = Δ+ O(log k)$. Finally, we define a more general notion of the diagonal Frobenius number for slacks $F_{ ext{slack}}(A)$, which is a generalization of $F_{ ext{diag}}(A)$ for canonical-form systems, like $A x leq b$. All the proofs are mainly done with respect to $F_{ ext{slack}}(A)$. The proof technique uses some properties of the Gomory's corner polyhedron relaxation and tools from discrepancy theory.

Problem

• Computing diagonal Frobenius number for integer matrices
• Finding integer solutions to linear Diophantine equations
• Generalizing Frobenius numbers for slack systems

Innovation

• Diagonal Frobenius number via Gomory relaxation
• Polynomial-time algorithm for integer solutions
• Generalization to slack Frobenius number