[Paper Review] Independent sets and cuts in large-girth regular graphs

Endre Csóka|arXiv (Cornell University)|Feb 8, 2016

Limits and Structures in Graph Theory10 references17 citations

TL;DR

This paper presents a local algorithm that constructs independent sets of expected size 0.44533n in large-girth 3-regular graphs and 0.40407n in large-girth 4-regular graphs, improving the best-known lower bounds. It also constructs a cut of size 1.34105n in 3-regular graphs using a refined application of the Hoppen-Wormald technique with additional structural observations, significantly reducing the gap between known upper and lower bounds.

ABSTRACT

We present a local algorithm producing an independent set of expected size $0.44533n$ on large-girth 3-regular graphs and $0.40407n$ on large-girth 4-regular graphs. We also construct a cut (or bisection or bipartite subgraph) with $1.34105n$ edges on large-girth 3-regular graphs. These decrease the gaps between the best known upper and lower bounds from $0.0178$ to $0.01$, from $0.0242$ to $0.0123$ and from $0.0724$ to $0.0616$, respectively. We are using local algorithms, therefore, the method also provides upper bounds for the fractional coloring numbers of $1 / 0.44533 \approx 2.24554$ and $1 / 0.40407 \approx 2.4748$ and fractional edge coloring number $1.5 / 1.34105 \approx 1.1185$. Our algorithms are applications of the technique introduced by Hoppen and Wormald.

Motivation & Objective

• To close the gap between known upper and lower bounds for independence ratio and cut size in large-girth regular graphs.
• To develop a constructive, local algorithm applicable to all large-girth regular graphs, not just random ones.
• To improve the best-known lower bounds for independence ratio in 3-regular and 4-regular graphs.
• To enhance the bound for the maximum cut size in 3-regular graphs using a refined version of the Hoppen-Wormald method.
• To derive new fractional coloring and edge coloring bounds from the improved independent set and cut results.

Proposed method

• The paper uses the configuration model to simultaneously construct random regular graphs and independent sets via a local, constant-time distributed algorithm.
• It applies a novel transformation rule to contract paths of three vertices with specific color patterns, preserving independence and improving expected size.
• A color-flipping mechanism is used on white-vertices to maintain symmetry and increase good edges (in the cut) during the process.
• The method involves solving a system of differential equations describing component dynamics in the survival graph, tracking types like [R], [G], [W], and [RR].
• Observation 3 allows replacing three red vertices with a single red vertex, increasing good edges by 3 and bad edges by 1, improving cut size.
• The algorithm uses priority rules for processing components and queries neighbors when needed, maintaining symmetry between red and green colors.

Experimental results

Research questions

• RQ1 Can a local algorithm achieve a higher independence ratio in large-girth 3-regular graphs than previously known bounds?
• RQ2 What is the maximum cut size achievable in large-girth 3-regular graphs using a constructive local method?
• RQ3 How much can the gap between upper and lower bounds for independence ratio and cut size be reduced in 3- and 4-regular graphs?
• RQ4 Can the Hoppen-Wormald technique be enhanced with structural observations to improve bounds further?
• RQ5 What are the resulting fractional coloring and edge coloring bounds derived from improved independent set and cut constructions?

Key findings

• The algorithm achieves an expected independent set size of 0.44533n in large-girth 3-regular graphs, improving upon the prior best of 0.43757.
• For 4-regular graphs, the method yields an independent set of expected size 0.40407n, improving the prior best of 0.39213.
• A cut of size 1.34105n is constructed in 3-regular graphs, reducing the gap between known bounds from 0.0724 to 0.0616.
• The fractional chromatic number is upper bounded at approximately 2.24554 for 3-regular graphs and 2.4748 for 4-regular graphs.
• The fractional edge coloring number is bounded at approximately 1.1185 for 3-regular graphs.
• The results are derived from a numerically stable computation based on a system of differential equations modeling component evolution in the survival graph.