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An $11/6$-Approximation Algorithm for Vertex Cover on String Graphs

Édouard Bonnet, Paweł Rzążewski

Abstract

We present a 1.8334-approximation algorithm for Vertex Cover on string graphs given with a representation, which takes polynomial time in the size of the representation; the exact approximation factor is $11/6$. Recently, the barrier of 2 was broken by Lokshtanov et al. [SoGC '24] with a 1.9999-approximation algorithm. Thus we increase by three orders of magnitude the distance of the approximation ratio to the trivial bound of 2. Our algorithm is very simple. The intricacies reside in its analysis, where we mainly establish that string graphs without odd cycles of length at most 11 are 8-colorable. Previously, Chudnovsky, Scott, and Seymour [JCTB '21] showed that string graphs without odd cycles of length at most 7 are 80-colorable, and string graphs without odd cycles of length at most 5 have bounded chromatic number.

An $11/6$-Approximation Algorithm for Vertex Cover on String Graphs

Abstract

We present a 1.8334-approximation algorithm for Vertex Cover on string graphs given with a representation, which takes polynomial time in the size of the representation; the exact approximation factor is . Recently, the barrier of 2 was broken by Lokshtanov et al. [SoGC '24] with a 1.9999-approximation algorithm. Thus we increase by three orders of magnitude the distance of the approximation ratio to the trivial bound of 2. Our algorithm is very simple. The intricacies reside in its analysis, where we mainly establish that string graphs without odd cycles of length at most 11 are 8-colorable. Previously, Chudnovsky, Scott, and Seymour [JCTB '21] showed that string graphs without odd cycles of length at most 7 are 80-colorable, and string graphs without odd cycles of length at most 5 have bounded chromatic number.
Paper Structure (14 sections, 13 theorems, 2 figures)

This paper contains 14 sections, 13 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Outline of the proof of \ref{['thm:coloring']}.
  3. Chromatic number of intersection graphs without short (odd) cycles.
  4. Limits of the method.
  5. Further discussion and questions.
  6. Definitions and preparatory lemmas
  7. Graphs and odd cycles
  8. String representations, and some useful notions and properties
  9. String graphs of odd girth larger than $11$ are $8$-colorable
  10. Definition of the topological disk $\bm{D}$.
  11. Definition of the auxiliary graph $\bm{H}$.
  12. $G[X_k]$ is bipartite
  13. 8-coloring algorithm
  14. Approximation algorithm for Vertex Cover

Key Result

Theorem 1

Vertex Cover admits an $\frac{11}{6}$-approximation algorithm in string graphs given with a representation, whose running time is polynomial in the size of a representation.

Figures (2)

  • Figure 1: Left: 5-vertex path as the intersection graph of strings, where the first string of the path contains point $a$, and the last string of the path contains point $b$. Right: illustration of $s[a,P,b]$.
  • Figure 3: The string $w_{i'} \in W$ with several neighbors in $V(C_p)$, and the new odd induced cycle $C_{p+1}$ obtained by \ref{['lem:odd-cheese']}. \ref{['lem:cell']} will then locate $s(w)$ as enclosed by $x, w_{i'}$ and some (here, three) strings of $V$. For legibility, a string may be labeled by its corresponding vertex.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 4
  • Conjecture 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • Lemma 11
  • Lemma 12
  • ...and 4 more