近似容量受限最小闭系统的Chvátal--Gomory闭包
Approximating the Chvátal--Gomory Closure of Capacity-Bounded Min-Closed Systems
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中文总结 AI 辅助
本文针对容量受限最小闭系统，证明了常数次平方和松弛可为其第一CG闭包上的线性目标最大化提供PTAS，扩展了混合符号约束下的近似保证。
中文摘要 AI 辅助
在二元整数线性规划的{0, 1/2}秩1 Chvátal-Gomory (CG)闭包上进行优化是NP难的。虽然对于单调打包和覆盖公式已建立了多项式时间近似方案(PTAS)，但将这些保证扩展到混合符号变体仍然是一个开放的挑战。在本文中，我们研究了k-松弛受限（容量受限）最小闭系统的CG闭包的可近似性，这是一种具有混合符号约束的广义打包公式，扩展了加权布尔Horn逻辑。在这些系统中，一个常数上界$k \ge 1$限制了每个约束的负惩罚系数$q$与右侧容量$b$之间的比率，强制满足$q \le k \cdot b$。我们证明了对于任意常数整数$f \ge 2$，一个常数次平方和松弛对于在由乘数$\{0\}\cup[\tfrac1f,1]$生成的第一CG闭包上最大化线性目标产生一个PTAS。该闭包包含在{0, 1/2}秩1 CG闭包中。
英文摘要
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 $f \ge 2$.
发表机构
- Università della Svizzera italiana（瑞士意大利语大学）
- Istituto Dalle Molle di studi sull’intelligenza artificiale (IDSIA USI-SUPSI)（达莱莫勒人工智能研究研究所（IDSIA USI-SUPSI））
- Scuola universitaria professionale della Svizzera italiana（瑞士意大利语专业大学）
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