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XALP-completeness of Parameterized Problems on Planar Graphs

Hans L. Bodlaender, Krisztina Szilágyi

TL;DR

The paper investigates the hardness landscape of parameterized problems on planar graphs under outerplanarity, treewidth, and pathwidth within the XNLP/XALP framework. It develops a rich set of gadget-based reductions from All-or-Nothing Flow and Binary CSP to derive XALP-hardness results for several natural problems on planar graphs parameterized by outerplanarity, including All-or-Nothing Flow, Target Outdegree Orientation, capacitated domination and vertex cover variants, f-/k-Domination, and Target Set Selection; it also establishes XNLP-hardness for Scattered Set (parameterized by pathwidth) and XNLP-completeness for Binary CSP on $k imes n$ grids. The main technique combines planarity-preserving reductions, crossover gadgets, and Sidon-set encodings to simulate color choices and flow while controlling the width-parameters; these results show that moving from treewidth to outerplanarity does not inherently ease the problems studied. Collectively, the results map a broad hardness landscape on planar graphs under outerplanarity and supply a framework for future work on related planarity-parameterized problems.

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.

XALP-completeness of Parameterized Problems on Planar Graphs

TL;DR

The paper investigates the hardness landscape of parameterized problems on planar graphs under outerplanarity, treewidth, and pathwidth within the XNLP/XALP framework. It develops a rich set of gadget-based reductions from All-or-Nothing Flow and Binary CSP to derive XALP-hardness results for several natural problems on planar graphs parameterized by outerplanarity, including All-or-Nothing Flow, Target Outdegree Orientation, capacitated domination and vertex cover variants, f-/k-Domination, and Target Set Selection; it also establishes XNLP-hardness for Scattered Set (parameterized by pathwidth) and XNLP-completeness for Binary CSP on grids. The main technique combines planarity-preserving reductions, crossover gadgets, and Sidon-set encodings to simulate color choices and flow while controlling the width-parameters; these results show that moving from treewidth to outerplanarity does not inherently ease the problems studied. Collectively, the results map a broad hardness landscape on planar graphs under outerplanarity and supply a framework for future work on related planarity-parameterized problems.

Abstract

The class XNLP consists of (parameterized) problems that can be solved nondeterministically in time and space, where is the size of the input instance and 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.
Paper Structure (21 sections, 21 theorems, 11 figures, 1 table)

This paper contains 21 sections, 21 theorems, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our results.
  3. Organization.
  4. Definitions and Notation
  5. Graph notions
  6. The classes XNLP and XALP
  7. Binary CSP
  8. XNLP-completeness for $k\times n$-grids
  9. XALP-completeness parameterized by outerplanarity
  10. Scattered Set
  11. All-or-Nothing Flow
  12. Gadgets.
  13. Reductions from All-or-Nothing Flow
  14. All-or-Nothing Flow with Small Arc Capacities
  15. Target Outdegree Orientation
  16. ...and 6 more sections

Key Result

lemma 1

A graph $G$ has pathwidth $k$ if and only if it is a subgraph of an interval graph $H$ such that $\omega(H)=k+1$.

Figures (11)

  • Figure 1:
  • Figure 2: Representation of nodes: Introduce $u$, Forget $x$, Add-Edge $vw$, Swap $vw$, Join.
  • Figure 3:
  • Figure 4: (a) The XOR-gadget for a vertex $v$ and its representation. (b) The XOR-gadget for an edge $\{v,w\}$ and its representation. (c) Changing a XOR-gadget to the $s$-$t$-gadget. $S(v)+1$ denotes the set $\{x+1~|~x\in S(v)\}$.
  • Figure 5: The construction for a vertex with neighbours
  • ...and 6 more figures

Theorems & Definitions (43)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • lemma 1: bodlaender1998partial
  • theorem 1
  • proof
  • theorem 2
  • proof
  • corollary 1
  • ...and 33 more