Exact Algorithms for Clustered Planarity with Linear Saturators

TL;DR

This paper introduces exact algorithms for Clustered Planarity with Linear Saturators, achieving subexponential time solutions and kernelization, while establishing NP-hardness even with few clusters, advancing understanding of the problem's complexity.

Contribution

It provides the first subexponential algorithms, kernelization results based on vertex cover, and NP-hardness proofs for the problem with limited clusters.

Findings

Algorithms run in $2^{O(n)}$ time for general cases.

Subexponential $2^{O(\sqrt{n}\log n)}$ time algorithm for connected graphs with fixed embedding.

NP-hardness persists even with at most 3 clusters.

Abstract

We study Clustered Planarity with Linear Saturators, which is the problem of augmenting an $n$-vertex planar graph whose vertices are partitioned into independent sets (called clusters) with paths - one for each cluster - that connect all the vertices in each cluster while maintaining planarity. We show that the problem can be solved in time $2^{O(n)}$ for both the variable and fixed embedding case. Moreover, we show that it can be solved in subexponential time $2^{O(\sqrt{n}\log n)}$ in the fixed embedding case if additionally the input graph is connected. The latter time complexity is tight under the Exponential-Time Hypothesis. We also show that $n$ can be replaced with the vertex cover number of the input graph by providing a linear (resp. polynomial) kernel for the variable-embedding (resp. fixed-embedding) case; these results contrast the NP-hardness of the problem on graphs of bounded treewidth (and even on trees). Finally, we complement known lower bounds for the problem by showing that Clustered Planarity with Linear Saturators is NP-hard even when the number of clusters is at most $3$, thus excluding the algorithmic use of the number of clusters as a parameter.