In the weighted minimum strongly connected spanning subgraph (WMSCSS ) problem we must purchase a minimum-cost strongly connected spanning subgraph of a digraph. We show that half-integral linear program (LP) solutions for WMSCSS can be efficiently rounded to integral solutions at a multiplicative 1.5 cost. This rounding matches a known 1.5 integrality gap lower bound for a half-integral instance. More generally, we show that LP solutions whose non-zero entries are at least a value $f>0$ can be rounded at a multiplicative cost of $2-f$.

中文翻译：

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半积分加权最小强连通跨子图的最佳舍入

在加权最小强连通跨子图（WMSCSS）问题中，我们必须购买图的最小成本强连通跨子图。我们展示了WMSCSS的半积分线性程序（LP）解决方案可以有效地四舍五入为整数解决方案，而费用却是1.5倍。此舍入匹配半积分实例的已知1.5积分间隙下限。更笼统地说，我们证明了非零项至少为一个值的LP解$F>0$ 可以四舍五入的方式四舍五入 $2-F$。