Approximating the Chvátal--Gomory Closure of Capacity-Bounded Min-Closed Systems

2026-09-28

AI Summary

This study addresses the NP-hardness of optimizing over the Chvátal-Gomory (CG) closure under mixed sign constraints, where existing polynomial-time approximation schemes (PTAS) are restricted to monotone formulas. By integrating Sum-of-Squares (SOS) relaxations with CG cutting plane theory, this work proposes constant-degree SOS relaxations for maximizing linear objectives over the CG closure in capacity-constrained minimum closure systems. The primary contribution is the first extension of PTAS to generalized packing models with negative coefficients and weighted Boolean Horn logic. Furthermore, it proves that for any constant f ≥ 2, a PTAS based on constant-degree SOS relaxations exists for CG closures generated by specific multipliers, thereby overcoming previous monotonicity limitations.

Abstract

Optimizing over the {0, 1/2} rank-1 Chvàtal-Gomory (CG) closure of a binary integer linear program is NP-hard. While polynomial-time approximation schemes (PTAS) are established for monotone packing and covering formulations, extending these guarantees to mixed-sign variants remains an open challenge. In this paper, we study the approximability of the CG closure for k-slack bounded (capacity-bounded) min-closed systems, a generalized packing formulation with mixed-sign constraints that extends weighted Boolean Horn logic. In these systems, a constant upper bound $k \ge 1$ bounds the ratio between the negative penalty coefficient $q$ and the right-hand side capacity $b$ of every constraint, enforcing $q \le k \cdot b$. We prove that a constant-degree sum-of-squares relaxation yields a PTAS for maximizing linear objectives over the first CG closure generated by multipliers in $\{0\}\cup[\tfrac1f,1]$, for any constant integer $f \ge 2$. This closure is contained in the {0, 1/2} rank-1 CG closure.Optimizing over the {0, 1/2} rank-1 CG closure of a binary integer linear program is NP-hard. While PTASes are established for monotone packing and covering formulations, extending these guarantees to mixed-sign variants remains an open challenge. In this paper, we study the approximability of the CG closure for \emph{$k$-slack bounded (capacity-bounded) min-closed systems}, a generalized packing formulation with mixed-sign constraints that extends weighted Boolean Horn logic. In these systems, a constant upper bound $k \ge 1$ bounds the ratio between the negative penalty coefficient $q$ and the right-hand side capacity $b$ of every constraint, enforcing $q \le k \cdot b$. We prove that a constant-degree sum-of-squares relaxation yields a PTAS for maximizing linear objectives over the first CG closure generated by multipliers in $\{0\}\cup[\tfrac1f,1]$, for any constant integer $f \ge 2$.

Problem

• Chvátal-Gomory closure
• binary integer linear program
• approximability
• capacity-bounded min-closed systems
• mixed-sign constraints

Innovation

• Chvátal-Gomory closure
• Sum-of-Squares relaxation
• Polynomial-time approximation scheme
• Min-closed systems
• Mixed-sign constraints