# An Optimal Rounding for Half-Integral Weighted Minimum Strongly Connected Spanning Subgraph

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.

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