Plane Strong Connectivity Augmentation

2025-12-19

AI Summary

This paper studies the strong connectivity augmentation problem for planar digraphs: given a planar digraph (D) and an integer (k), does there exist an arc set (X) of size at most (k) such that (D + X) remains planar and strongly connected? This is an NP-hard budgeted connectivity augmentation problem. We present the first fixed-parameter tractable (FPT) algorithm for this problem under planarity constraints. Our method introduces a face-level structural domination theory and designs a branching strategy based on face decomposition; reduces the problem to Minimum Dijoin; and derandomizes the Monte-Carlo reduction. The resulting deterministic FPT algorithm runs in time (2^{O(k)} n^{O(1)}). This breakthrough overcomes a fundamental barrier in designing efficient connectivity augmentation algorithms for planar digraphs, establishing the first FPT result for strong connectivity augmentation under planarity.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Deterministic Planning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• Augmenting plane oriented graphs to achieve strong connectivity
• Determining NP-hardness of plane strong connectivity augmentation
• Developing FPT algorithm for budgeted plane connectivity augmentation

Innovation

• Fixed-parameter tractable algorithm for plane strong connectivity augmentation
• Face-wise branching with bounded partial solutions per face
• Derandomized Monte-Carlo reduction to Minimum Dijoin problem