FPT Algorithm for PSCA: Planar Augmentation

• The paper introduces an FPT algorithm for PSCA that augments plane oriented graphs with at most k arcs to achieve strong connectivity while retaining planarity.
• It employs structural sparsification via face decomposition and classification, reducing the search space and ensuring branching over O(2^(O(k))) configurations.
• The approach further reduces simple face problems to a Minimum Dijoin instance using derandomized universal sets, achieving an overall runtime of O(2^(O(k)) n^(O(1))).

The Fixed-Parameter Tractable (FPT) algorithm for Plane Strong Connectivity Augmentation (PSCA) addresses the problem of augmenting an oriented plane graph with a limited number of additional arcs to achieve strong connectivity while maintaining planarity and orientation constraints. This problem parametrizes by the augmentation budget $k$, and the primary contribution is an $O(2^{O(k)} n^{O(1)})$-time FPT algorithm—the first such tractable result for a strongly planar-constrained connectivity augmentation problem (Bessy et al., 19 Dec 2025).

1. Problem Statement and Formulation

Plane Strong Connectivity Augmentation (PSCA) is defined as follows: Given a plane oriented graph $D = (V, A)$—that is, a simple directed graph with a fixed crossing-free embedding and no 2-cycles—and an integer $k \geq 0$, the task is to determine whether there exists a set $X \subseteq V \times V$ of at most $k$ new arcs, each drawn in a single face of $D$, such that the augmented graph $D + X$ remains oriented (no bidirected pairs), planar (respects the given embedding), and is strongly connected.

Planarity is relative to the embedding: only arcs embeddable in a particular face are eligible. Oriented means that no two vertices have arcs in both directions and no new digons are introduced. This strict constraint ensures the augmentation respects the input’s topological and directionality requirements.

2. Structural Sparsification via Face Decomposition

Central to the algorithm is the structural sparsification lemma: For any face $F$ of $D$ and an augmentation budget $k$, there exists at most $2^{O(k)}$ "dominating" augmentation sets $X \subseteq V(F)^2$ of size at most $k$ such that every other completion in $F$ is dominated by one of these. Here, domination means that the associated partition of $V(F)$ into strongly connected components in $D+X$ is a refinement of that induced by any other augmentation in this family. As a consequence, face-wise branching in the dynamic programming algorithm only needs to retain these representatives per face for further consideration.

This sparsification enables the algorithm to reduce the combinatorial search space by several orders of magnitude, bounding the otherwise exponential increase with respect to $k$. The approach is reminiscent of representative set constructions in parameterized complexity.

3. Face Classification and Branching Scheme

The algorithm distinguishes two face types based on local sources and sinks along their boundary. For a face $F$, a vertex $v$ is a local source/sink if its incident boundary arcs in $F$ are directed out of/into $v$, respectively. The count $lt(F)$ of local terminals (sources and sinks) around $F$—with sources and sinks alternating—determines the classification:

• Simple face: $lt(F) = 2$
• Alternating face: $lt(F) \geq 4$

The total number of local terminals across all alternating faces is $O(k)$ in any positive instance (from an Eulerian argument), implying that there are $O(k)$ such faces overall. For every alternating face, the algorithm branches over all $2^{O(lt(F))}$ supported completions. As $\sum_F lt(F) = O(k)$, the total number of branches is $2^{O(k)}$.

For each branch, the corresponding set $Y_{\text{AF}}$ of arcs augmenting the alternating faces is fixed, reducing the instance to a residual subproblem on simple faces with a diminished budget $k' = k - |Y_{\text{AF}}|$.

4. Reduction to the Minimum Dijoin Problem and Derandomization

The portion of the graph consisting only of simple faces after completion of the alternating faces can be handled by further reduction. For each simple face, minimal augmenting completions are enumerated (constant number per face), and a selection of at most $k'$ is needed to ensure strong connectivity.

A randomized process selects a representative for each simple face, yielding, after at most $2^{O(k')}$ iterations, a candidate set $U$ of new arcs to consider. To evaluate whether $D'$ (the graph after augmenting alternating faces) plus $U$ can be made strongly connected within budget, the algorithm constructs an auxiliary digraph $G'$ by $(k'+1)$-subdivision and attachment of gadgets for each chosen arc.

The core of the solution is solving the Minimum Dijoin problem in $G'$, which is polynomial-time tractable by Frank’s algorithm. This reduction yields a one-sided Monte Carlo algorithm: false positives are precluded, while any existing solution is found with high probability.

Derandomization employs universal sets: a family $H$ of functions from the simple faces to their options, such that for any subset of at most $k'$ faces and selection of completions, some $h \in H$ achieves that selection. Known constructions provide sets of requisite size, ensuring overall runtime remains $2^{O(k)} n^{O(1)}$.

5. Algorithmic Workflow and Complexity Analysis

The full PSCA FPT algorithm proceeds as follows:

1. Face classification: Partition faces into simple and alternating, and reject if $\sum lt(F) > 8k$.
2. Branching over alternating faces: For each choice of supported completions in alternating faces, update the digraph and budget.
3. Simple faces subproblem: For each branch, enumerate all combinations over supported completions in simple faces using a derandomized universal set.
4. Global solution for connectivity: For each such combination, construct the auxiliary digraph and invoke the Minimum Dijoin algorithm.
5. Reassembly and verification: If any branch returns YES, accept; else, return NO.

The branching over alternating faces and use of universal sets ensures that the total number of calls to the polynomial-time subroutine is $2^{O(k)}$. All other operations—face classification, local completion enumeration, construction of the auxiliary graphs—are performed in polynomial time.

Overall, the algorithm solves PSCA in $O(2^{c k} n^d)$ time for absolute constants $c, d$, i.e., FPT in budget $k$ (Bessy et al., 19 Dec 2025).

6. Extensions and Open Questions

The presented method generalizes to oriented graphs embedded on any fixed-genus surface; genus $g$ admits FPT parameterization in $k$ for each fixed $g$. For arbitrary planar graphs without a fixed embedding ("planar" rather than "plane" inputs), the complexity remains unresolved.

In the undirected setting, the analogue—minimally augmenting to biconnectivity while preserving planarity—is NP-hard; whether an FPT algorithm exists when parameterized by the augmentation budget is an open research question. The face-wise branching and reduction to global augmentation via Minimum Dijoin may transfer to related planar augmentation scenarios.

A plausible implication is that the decomposition-branching-plus-global-augmentation architecture forms a robust template for connectivity augmentation in combinatorially and topologically constrained settings.

7. Summary Table: Core Components and Guarantees

Component	Description	Complexity Bound
Structural lemma	$2^{O(k)}$ face completions suffice per face	Existence guarantee
Face-wise branching	Branch over $2^{O(k)}$ combinations for alternating faces	$2^{O(k)}$ branches
Simple faces subproblem	Universal set + polytimes Minimum Dijoin reductions	$2^{O(k)} n^{O(1)}$
Final runtime	Total time for PSCA solution	$2^{O(k)} n^{O(1)}$

The FPT algorithm for PSCA, via structural decomposition and global reductions, resolves the fixed-parameter tractability frontier for budgeted strong connectivity augmentation in the plane, and opens avenues for further parameterized study in topologically constrained graph augmentation (Bessy et al., 19 Dec 2025).