Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximation Ratio of the Min-Degree Greedy Algorithm for Maximum Independent Set on Interval and Chordal Graphs

Steven Chaplick, Martin Frohn, Steven Kelk, Johann Lottermoser, Matus Mihalak

TL;DR

The paper investigates the MIS problem under the min-degree greedy algorithm with adversarial tie-breaking on interval and chordal graphs. It proves a $1/2$-approximation for chordal graphs and a $2/3$-approximation for interval graphs, with tightness in both cases, contrasting these constants with the general $3/\big(\Delta+2\big)$ bound for arbitrary graphs. The analysis employs move-based reasoning and tree-decomposition concepts (maximal cliques and, for interval graphs, path decompositions) and provides tightness constructions, including unit-interval graphs of maximum degree $3$. It also shows that these constant-factor guarantees do not extend to natural generalizations such as permutation and 2-track interval graphs, and discusses open questions and the potential role of tie-breaking advice.

Abstract

In this article we prove that the minimum-degree greedy algorithm, with adversarial tie-breaking, is a $(2/3)$-approximation for the Maximum Independent Set problem on interval graphs. We show that this is tight, even on unit interval graphs of maximum degree 3. We show that on chordal graphs, the greedy algorithm is a $(1/2)$-approximation and that this is again tight. These results contrast with the known (tight) approximation ratio of $\frac{3}{Δ+2}$ of the greedy algorithm for general graphs of maximum degree $Δ$.

Approximation Ratio of the Min-Degree Greedy Algorithm for Maximum Independent Set on Interval and Chordal Graphs

TL;DR

The paper investigates the MIS problem under the min-degree greedy algorithm with adversarial tie-breaking on interval and chordal graphs. It proves a -approximation for chordal graphs and a -approximation for interval graphs, with tightness in both cases, contrasting these constants with the general bound for arbitrary graphs. The analysis employs move-based reasoning and tree-decomposition concepts (maximal cliques and, for interval graphs, path decompositions) and provides tightness constructions, including unit-interval graphs of maximum degree . It also shows that these constant-factor guarantees do not extend to natural generalizations such as permutation and 2-track interval graphs, and discusses open questions and the potential role of tie-breaking advice.

Abstract

In this article we prove that the minimum-degree greedy algorithm, with adversarial tie-breaking, is a -approximation for the Maximum Independent Set problem on interval graphs. We show that this is tight, even on unit interval graphs of maximum degree 3. We show that on chordal graphs, the greedy algorithm is a -approximation and that this is again tight. These results contrast with the known (tight) approximation ratio of of the greedy algorithm for general graphs of maximum degree .
Paper Structure (8 sections, 9 theorems, 8 equations, 2 figures)

This paper contains 8 sections, 9 theorems, 8 equations, 2 figures.

Table of Contents

  1. Introduction and Preliminaries
  2. Background
  3. Further preliminaries
  4. Tight Bounds: Chordal and Interval Graphs
  5. Lower bounds on the approximation ratio
  6. Tightness
  7. Beyond Chordal and Interval Graphs: Perfect and 2-Track Interval Graphs
  8. Conclusion and Open Questions

Key Result

lemma 1.4

Let $G$ be a (not necessarily chordal) graph and let $v$ be the first vertex picked by Greedy. Suppose $v$ has $k \geq 2$ neighbours $u_1, \ldots, u_k$ that are pairwise non-adjacent, i.e. $\{u_1,\ldots,u_k\}$ is an independent set. Then each vertex $u_1, \ldots, u_k$ has at least $k-1$ neighbours t

Figures (2)

  • Figure 1: A schematic depiction of the situation in the proof of Lemma \ref{['lem:disconnect']}. The vertices $w_1, \ldots, w_k$ will be in separate connected components of $G \setminus N[v]$. The set $N[v]$ is equal to the union of all the bags in $\mathbb{T}_v$.
  • Figure 2: The tight construction for interval graphs when $k=3$. In the top part of the figure is the graph itself, in the bottom part is a representation of the interval graph using unit-length intervals. The grey vertices indicate those which Greedy might pick first.

Theorems & Definitions (19)

  • proof
  • lemma 1.4
  • proof
  • lemma 1.5
  • proof
  • lemma 1.6
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • ...and 9 more