Approximating Minimum Cost Connectivity Orientation and Augmentation

TL;DR

This paper presents a polynomial-time 6-approximation algorithm for a complex connectivity and orientation problem, introduces a novel LP formulation with partition constraints, and disproves a conjecture about integrality gaps.

Contribution

It develops a new approximation algorithm using an innovative LP approach and uncrossing technique, and shows the problem's complexity with orientation-dependent costs.

Findings

Provides a 6-approximation algorithm for the problem.

Introduces an alternative LP with partition and co-partition constraints.

Disproves the conjecture on bounded integrality gap for fixed k.

Abstract

We investigate problems addressing combined connectivity augmentation and orientations settings. We give a polynomial-time 6-approximation algorithm for finding a minimum cost subgraph of an undirected graph $G$ that admits an orientation covering a nonnegative crossing $G$-supermodular demand function, as defined by Frank. An important example is $(k,\ell)$-edge-connectivity, a common generalization of global and rooted edge-connectivity. Our algorithm is based on a non-standard application of the iterative rounding method. We observe that the standard linear program with cut constraints is not amenable and use an alternative linear program with partition and co-partition constraints instead. The proof requires a new type of uncrossing technique on partitions and co-partitions. We also consider the problem setting when the cost of an edge can be different for the two possible orientations. The problem becomes substantially more difficult already for the simpler requirement of $k$-edge-connectivity. Khanna, Naor, and Shepherd showed that the integrality gap of the natural linear program is at most $4$ when $k=1$ and conjectured that it is constant for all fixed $k$. We disprove this conjecture by showing an $\Omega(|V|)$ integrality gap even when $k=2$.