 
Summary: On constant time approximation of parameters
of bounded degree graphs
Noga Alon
August 22, 2010
Abstract
How well can the maximum size of an independent set, or the minimum size of a dominating
set of a graph in which all degrees are at most d be approximated by a randomized constant time
algorithm ? Motivated by results and questions of Nguyen and Onak, and of Parnas, Ron and
Trevisan, we show that the best approximation ratio that can be achieved for the first question
(independence number) is between (d/ log d) and O(d log log d/ log d), whereas the answer to the
second (domination number) is (1 + o(1)) ln d.
1 Introduction
The question of identifying the properties of bounded degree graphs in the model of [7] that can be
tested efficiently, is that of recognizing the properties that are local in nature. These are properties for
which the local structure of the graph supplies meaningful information about the global property. A
related problem deals with efficient approximation algorithms for graph parameters, like the indepen
dence number, or the domination number of a given bounded degree graph. The question here is to
decide how well we can approximate these quantities by observing the local structure of the graph. In
this short paper we discuss several problems of this type, continuing the work in several earlier papers
including [13] and [12] on related questions.
