Dual Half-integrality for Uncrossable Cut Cover and its Application to Maximum Half-Integral Flow

TL;DR

This paper introduces a half-integral dual approach for the 2-edge connectivity augmentation problem, leading to a tight 2-approximate max-half-integral-flow min-multicut theorem, with implications for planar supply-demand routing.

Contribution

It extends the primal-dual algorithm to produce a half-integral dual solution for integral weights, establishing a new approximation bound for the problem.

Findings

Half-integral dual solutions can be obtained for integral weights.

The approach yields a tight 2-approximate max-half-integral-flow min-multicut theorem.

Connections to planar supply-demand routing are established.

Abstract

Given an edge weighted graph and a forest $F$, the $\textit{2-edge connectivity augmentation problem}$ is to pick a minimum weighted set of edges, $E'$, such that every connected component of $E'\cup F$ is 2-edge connected. Williamson et al. gave a 2-approximation algorithm (WGMV) for this problem using the primal-dual schema. We show that when edge weights are integral, the WGMV procedure can be modified to obtain a half-integral dual. The 2-edge connectivity augmentation problem has an interesting connection to routing flow in graphs where the union of supply and demand is planar. The half-integrality of the dual leads to a tight 2-approximate max-half-integral-flow min-multicut theorem.