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Local Constant Approximation for Dominating Set on Graphs Excluding Large Minors

Marthe Bonamy, Cyril Gavoille, Timothé Picavet, Alexandra Wesolek

TL;DR

This work advances distributed computation for classical graph problems by showing that large excluded minors do not preclude constant‑round, constant‑factor approximations. Focusing on $K_{2,t}$‑minor‑free graphs, the authors design a deterministic LOCAL algorithm that achieves a 50‑approximation for Minimum Dominating Set (and Minimum Vertex Cover) in $O_t(1)$ rounds, and also give an $O(1)$‑round $(2t+1)$‑approximation; they further present a 3‑round $(2t-1)$‑approximation. The core technique hinges on bounding the influence of local structures via asymptotic dimension, allowing a reduction to small bounded‑radius components that can be solved by brute force. The results demonstrate that substantial excluded minors can yield fast, scalable, and constant‑factor distributed algorithms, and they introduce SPQR trees and interesting 2‑cut forests as key tools in the analysis. Overall, the paper contributes new theoretical tools and a concrete algorithmic framework for translating local properties into global approximations in classes with bounded asymptotic dimension.

Abstract

We show that graphs excluding $K_{2,t}$ as a minor admit a $f(t)$-round $50$-approximation deterministic distributed algorithm for Minimum Dominating Set. The result extends to Minimum Vertex Cover. Though fast and approximate distributed algorithms for such problems were already known for $H$-minor-free graphs, all of them have an approximation ratio depending on the size of $H$. To the best of our knowledge, this is the first example of a large non-trivial excluded minor leading to fast and constant-approximation distributed algorithms, where the ratio is independent of the size of $H$. A new key ingredient in the analysis of these distributed algorithms is the use of asymptotic dimension.

Local Constant Approximation for Dominating Set on Graphs Excluding Large Minors

TL;DR

This work advances distributed computation for classical graph problems by showing that large excluded minors do not preclude constant‑round, constant‑factor approximations. Focusing on ‑minor‑free graphs, the authors design a deterministic LOCAL algorithm that achieves a 50‑approximation for Minimum Dominating Set (and Minimum Vertex Cover) in rounds, and also give an ‑round ‑approximation; they further present a 3‑round ‑approximation. The core technique hinges on bounding the influence of local structures via asymptotic dimension, allowing a reduction to small bounded‑radius components that can be solved by brute force. The results demonstrate that substantial excluded minors can yield fast, scalable, and constant‑factor distributed algorithms, and they introduce SPQR trees and interesting 2‑cut forests as key tools in the analysis. Overall, the paper contributes new theoretical tools and a concrete algorithmic framework for translating local properties into global approximations in classes with bounded asymptotic dimension.

Abstract

We show that graphs excluding as a minor admit a -round -approximation deterministic distributed algorithm for Minimum Dominating Set. The result extends to Minimum Vertex Cover. Though fast and approximate distributed algorithms for such problems were already known for -minor-free graphs, all of them have an approximation ratio depending on the size of . To the best of our knowledge, this is the first example of a large non-trivial excluded minor leading to fast and constant-approximation distributed algorithms, where the ratio is independent of the size of . A new key ingredient in the analysis of these distributed algorithms is the use of asymptotic dimension.

Paper Structure

This paper contains 28 sections, 17 theorems, 7 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. The LOCAL Model.
  3. Fast Algorithms.
  4. Constant-Round Algorithms.
  5. $H$-minor-free With Large $H$.
  6. Our Contributions.
  7. Sketch of the Main Algorithm ( \ref{['TH:MAIN']}TH:MAIN).
  8. Preliminaries
  9. General Definitions.
  10. Local Connectivity.
  11. Asymptotic Dimension
  12. When Local Properties Can Replace Global Properties
  13. Bounding the Number of Local 1-cuts and 2-cuts
  14. Constant Approximation for Minimum Dominating Set
  15. Intuition and Explanation.
  16. ...and 13 more sections

Key Result

Proposition 3.1

Let $\mathcal{C}$ be a hereditary class of graphs and $k\geqslant 0$. Let $\mathcal{A}$ be a local algorithm with round complexity $r\geqslant 1$ and with the following property: for every $G\in\mathcal{C}$ and $S\subseteq V(G)$, $|\mathcal{A}(G)\cap S|\leqslant \alpha \cdot \mathop{\mathrm{MDS}}\no

Figures (2)

  • Figure 1: An explanation of the preprocessing procedure. Here the dashed squiggly edge represents the lack of path in $G[B]$.
  • Figure 2: The vertices at the top are in $A'$ and the vertices on bottom are in $B$. In the yellow region, the neighbors $a_i$ of $c$ in $A'$, which are all connected to a vertex $b_i \in C\setminus\{c\}$ in the red region. $c$ and the $b_i$ are all in the same connected component of $G[B]$.

Theorems & Definitions (28)

  • Definition 2.1: Local cut
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Claim 5.1
  • Lemma 5.2
  • ...and 18 more