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

TL;DR

This paper presents an efficient rounding technique for half-integral LP solutions in the weighted minimum strongly connected spanning subgraph problem, achieving a 1.5 approximation ratio that matches the known integrality gap lower bound.

Contribution

It introduces a novel rounding method for half-integral LP solutions in WMSCSS, ensuring a tight 1.5 approximation ratio and generalizing to solutions with larger non-zero entries.

Findings

Half-integral LP solutions can be rounded at a 1.5 cost.

The rounding matches the known 1.5 integrality gap lower bound.

Generalization to solutions with entries at least f, achieving a 2 - f approximation.

Abstract

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 $2 - f$.