Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact Algorithms for Distance to Unique Vertex Cover

Foivos Fioravantes, Dušan Knop, Nikolaos Melissinos, Michal Opler, Manolis Vasilakis

TL;DR

The paper studies MU-VC, the problem of deleting a modulator to enforce a unique minimum vertex cover, and shows it is $oldsymbol{ obreakspace}oldsymbol{A3_2^P}$-complete even for planar graphs of maximum degree $5$. It provides a linear-time algorithm for MU-VC on trees and extends the approach to width-based settings, yielding an $n^{O(2^w)}$-time DP for treewidth $w$, an $FPT$ algorithm when additionally parameterized by maximum degree $oldsymbol{ore}$, and an $XP$-time algorithm parameterized by clique-width $d$. Furthermore, adding the solution size $k$ as a parameter yields an $FPT$-time algorithm for the clique-width setting, while the problem remains hard in general. The work also clarifies differences with PAU-VC by presenting hardness results and constructing reductions that preserve the unique minimum vertex cover property under vertex deletions. Overall, the paper maps the precise complexity landscape of MU-VC across classic structural graph parameters and establishes both tight hardness and tractable algorithms under structured inputs, contributing to dataset construction and fixed-parameter algorithm design for uniqueness guarantees in vertex cover.

Abstract

Horiyama et al. (AAAI 2024) studied the problem of generating graph instances that possess a unique minimum vertex cover under specific conditions. Their approach involved pre-assigning certain vertices to be part of the solution or excluding them from it. Notably, for the \textsc{Vertex Cover} problem, pre-assigning a vertex is equivalent to removing it from the graph. Horiyama et al.~focused on maintaining the size of the minimum vertex cover after these modifications. In this work, we extend their study by relaxing this constraint: our goal is to ensure a unique minimum vertex cover, even if the removal of a vertex may not incur a decrease on the size of said cover. Surprisingly, our relaxation introduces significant theoretical challenges. We observe that the problem is $Σ^2_P$-complete, and remains so even for planar graphs of maximum degree 5. Nevertheless, we provide a linear time algorithm for trees, which is then further leveraged to show that MU-VC is in \textsf{FPT} when parameterized by the combination of treewidth and maximum degree. Finally, we show that MU-VC is in \textsf{XP} when parameterized by clique-width while it is fixed-parameter tractable (FPT) if we add the size of the solution as part of the parameter.

Exact Algorithms for Distance to Unique Vertex Cover

TL;DR

The paper studies MU-VC, the problem of deleting a modulator to enforce a unique minimum vertex cover, and shows it is -complete even for planar graphs of maximum degree . It provides a linear-time algorithm for MU-VC on trees and extends the approach to width-based settings, yielding an -time DP for treewidth , an algorithm when additionally parameterized by maximum degree , and an -time algorithm parameterized by clique-width . Furthermore, adding the solution size as a parameter yields an -time algorithm for the clique-width setting, while the problem remains hard in general. The work also clarifies differences with PAU-VC by presenting hardness results and constructing reductions that preserve the unique minimum vertex cover property under vertex deletions. Overall, the paper maps the precise complexity landscape of MU-VC across classic structural graph parameters and establishes both tight hardness and tractable algorithms under structured inputs, contributing to dataset construction and fixed-parameter algorithm design for uniqueness guarantees in vertex cover.

Abstract

Horiyama et al. (AAAI 2024) studied the problem of generating graph instances that possess a unique minimum vertex cover under specific conditions. Their approach involved pre-assigning certain vertices to be part of the solution or excluding them from it. Notably, for the \textsc{Vertex Cover} problem, pre-assigning a vertex is equivalent to removing it from the graph. Horiyama et al.~focused on maintaining the size of the minimum vertex cover after these modifications. In this work, we extend their study by relaxing this constraint: our goal is to ensure a unique minimum vertex cover, even if the removal of a vertex may not incur a decrease on the size of said cover. Surprisingly, our relaxation introduces significant theoretical challenges. We observe that the problem is -complete, and remains so even for planar graphs of maximum degree 5. Nevertheless, we provide a linear time algorithm for trees, which is then further leveraged to show that MU-VC is in \textsf{FPT} when parameterized by the combination of treewidth and maximum degree. Finally, we show that MU-VC is in \textsf{XP} when parameterized by clique-width while it is fixed-parameter tractable (FPT) if we add the size of the solution as part of the parameter.

Paper Structure

This paper contains 24 sections, 8 theorems, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our Contribution.
  3. Preliminaries
  4. Trees
  5. Polynomial algorithm
  6. Leaf node, $(T,r) = \mathsf{Leaf}(r)$.
  7. Join node, $(T,r) = \mathsf{Join}((T_1, r), (T_2, r))$.
  8. Extend node, $(T,r) = \mathsf{Extend}((T', r'), r)$.
  9. Time complexity.
  10. Linear algorithm
  11. Leaf node, $(T,r) = \mathsf{Leaf}(r)$.
  12. Join node, $(T,r) = \mathsf{Join}((T_1, r), (T_2, r))$.
  13. Extend node, $(T,r) = \mathsf{Extend}((T', r'), r)$.
  14. Treewidth
  15. Leaf node.
  16. ...and 9 more sections

Key Result

Theorem 1

For every $k \geq 3$, there exists a graph with minimum solutions $P$ and $M$ for PAU-VC and MU-VC respectively, where $|P| \ge k$ and $|M| = 2$.

Figures (3)

  • Figure 1: Left: a graph $G$ with minimum vertex cover of size 4. There are many such -- vertex $1$ and then any set of vertices of size $3$ that intersects the pairs $\{2,3\}$, $\{4,5\}$, and $\{6,7\}$. Middle: PAU-VC solution of size 3 (the only vertex missing at this point is $1$). Right: MU-VC solution (deleted) $8,9$, and the minimum size of a vertex cover is now 3 (i.e., 5 in total); namely, $2,4,6$ which is now unique.
  • Figure 2: The gadgets used in the proof of \ref{['thm:sigma_2P']}.
  • Figure 3: An example of the construction for a formula $\phi$ containing the two clauses $c_1=(x_1\lor x_2\lor \lnot y_1)$ and $c_2=(\lnot x_1 \lor y_1 \lor y_2)$. The bold (olive resp.) edges represent the edges added in the first (second resp.) step of the edge-adding procedure.

Theorems & Definitions (42)

  • Theorem 1
  • proof
  • Theorem 2
  • Claim 1
  • Claim 2
  • Claim 3
  • Claim 4
  • Claim 5
  • Claim 6
  • Claim 7
  • ...and 32 more