Plane Strong Connectivity Augmentation

TL;DR

This paper studies the complexity of augmenting plane oriented graphs to achieve strong connectivity, proving NP-hardness for the general case and providing the first fixed-parameter tractable algorithm for the budgeted version.

Contribution

It introduces the first FPT algorithm for the NP-hard Plane Strong Connectivity Augmentation problem, utilizing structural graph properties and derandomization techniques.

Findings

Deciding augmentation for strong connectivity is NP-hard.

The paper presents a fixed-parameter tractable algorithm for the budgeted problem.

Structural results limit the number of partial solutions per face.

Abstract

We investigate the problem of strong connectivity augmentation within plane oriented graphs. We show that deciding whether a plane oriented graph $D$ can be augmented with (any number of) arcs $X$ such that $D+X$ is strongly connected, but still plane and oriented, is NP-hard. This question becomes trivial within plane digraphs, like most connectivity augmentation problems without a budget constraint. The budgeted version, Plane Strong Connectivity Augmentation (PSCA) considers a plane oriented graph $D$ along with some integer $k$, and asks for an $X$ of size at most $k$ ensuring that $D+X$ is strongly connected, while remaining plane and oriented. Our main result is a fixed-parameter tractable algorithm for PSCA, running in time $2^{O(k)} n^{O(1)}$. The cornerstone of our procedure is a structural result showing that, for any fixed $k$, each face admits a bounded number of partial solutions "dominating" all others. Then, our algorithm for PSCA combines face-wise branching with a Monte-Carlo reduction to the polynomial Minimum Dijoin problem, which we derandomize. To the best of our knowledge, this is the first FPT algorithm for a (hard) connectivity augmentation problem constrained by planarity.