XALP-completeness of Parameterized Problems on Planar Graphs

TL;DR

This paper establishes the computational hardness of several parameterized graph problems on planar graphs within the classes XNLP and XALP, highlighting their complexity even under structural restrictions.

Contribution

It proves XALP- and XNLP-completeness for multiple problems on planar graphs parameterized by outerplanarity, treewidth, and pathwidth, extending previous complexity results.

Findings

Several problems are XALP-complete on planar graphs with respect to outerplanarity.

Scattered Set is XNLP-complete when parameterized by pathwidth.

The results strengthen the understanding of the complexity landscape of graph problems on restricted planar graph classes.

Abstract

The class XNLP consists of (parameterized) problems that can be solved nondeterministically in $f(k)n^{O(1)}$ time and $f(k)\log n$ space, where $n$ is the size of the input instance and $k$ the parameter. The class XALP consists of problems that can be solved in the above time and space with access to an additional stack. These two classes are a "natural home" for many standard graph problems and their generalizations. In this paper, we show the hardness of several problems on planar graphs, parameterized by outerplanarity, treewidth and pathwidth, thus strengthening several existing results. In particular, we show the XALP-completeness of the following problems parameterized by outerplanarity: All-or-Nothing Flow, Target Outdegree Orientation, Capacitated (Red-Blue) Dominating Set, Target Set Selections etc. We also show the XNLP-completeness of Scattered Set parameterized by pathwidth and XALP-completeness parameterized by treewidth and outerplanarity.