Abstract combinatorial programs and efficient property testers

Abstract

Property testing is a relaxation of classical decision problems which aims at distinguishing between functions having a predetermined property and functions being far from any function having the property. In this paper we present a novel framework for analyzing property testing algorithms with one-sided error. Our framework is based on a connection of property testing and a new class of problems which we call abstract combinatorial programs. We show that if the problem of testing a property can be reduced to an abstract combinatorial program of small dimension, then the property has an efficient tester. We apply our framework to a variety of classical combinatorial problems. Among others, we present efficient property testing algorithms for geometric clustering problems, the reversal distance problem, and graph and hypergraph coloring problems. We also prove that, informally, any hereditary graph property can be efficiently tested if and only if it can be reduced to an abstract combinatorial program of small size. Our framework allows us to analyze all our testers in a unified way and the obtained complexity bounds either match or improve the previously known bounds. We believe that our framework will help to better understand the structure of efficiently testable properties.