Plane Strong Connectivity Augmentation

TL;DR

<3-5 sentence high-level summary> The paper addresses the problem of making a plane oriented digraph strongly connected by adding arcs while preserving planarity and orientation, introducing Plane Strong Connectivity Augmentation (PSCA) and proving its NP-hardness for the unbudgeted version. It then establishes that the budgeted PSCA is fixed-parameter tractable, with an algorithm running in 2^{O(k)} n^{O(1)} time, by carefully decomposing the embedding into faces, classifying faces into simple and alternating types, and employing face-wise branching together with reductions to Minimum Dijoin (derandomized). A key technical contribution is a structural result bounding the number of dominating partial solutions in each face for fixed k, enabling tractable search despite the planar constraints. The paper also shows that the corresponding Directed-PSCA variant is FPT and that PSCA' (the unembellished variant without the planarity constraint on the result) is NP-hard via a linear reduction from Planar-3-SAT, underscoring the intrinsic complexity of planarity-aware augmentation problems.

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.

Figures (9)

• Figure 1: A simple face, with two local terminals and an alternating face with $22$ local terminals. Local sources are shown as green angles, while local sinks are the red angles. Note that the same vertex can appear in both local sinks, local sources and even non-terminal angles.
• Figure 2: A directed path $P$ from a local source $s$ to a local sink $t$ of a face, with arcs of a solution incident to it shown in blue. Removing the arcs incident to vertex $v$ yields several source and sink components (in green and red), one of which is a source containing both $u$ and $v$. Shifting their endpoints to $u,$ as shown in purple, yields a solution incident to fewer internal vertices of $P.$
• Figure 3: An instance of PSCA, with a non-trivial strong component in orange, and the oriented graph obtained by contracting it into a single vertex. The initial instance has a solution of size three, shown in blue, that is not preserved for the condensation, which is a negative instance.
• Figure 4: A face $F$ of $D$, with arcs of $D[V(F)]$ embedded outside $F$ shown in gray. The strong components of $D$ are shown as clouds in green (source), red (sink) and orange (non-trivial intermediate). Each consecutive intersection of these components with the boundary of $F$ forms a strong interval. Strong intervals are shown by thick lines, in green for local sources, red for local sinks, and orange otherwise.
• Figure 5: A literal gadget $L$, with internal sinks $b,l,r,$ bottom sources $bl,br$ and top sources $tl,tr.$

Theorems & Definitions (43)

• Theorem 1
• Theorem 2
• Conjecture 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 6
• Corollary 7
• Definition 8: Strong intervals
• Definition 9: Local terminals