Property testing and parameter testing for permutations

• Authors: Carlos Hoppen;Yoshiharu Kohayakawa;Carlos Gustavo Moreira;Rudini Menezes Sampaio
• Venue: SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
• Year: 2010

There has been great interest in deciding whether a combinatorial structure satisfies some property, or in estimating the value of some numerical function associated with this combinatorial structure, by considering only a randomly chosen substructure of sufficiently large, but constant size. These problems are called property testing and parameter testing, where a property or parameter is said to be testable if it can be estimated accurately in this way. The algorithmic appeal is evident, as, conditional on sampling, this leads to reliable constant-time randomized estimators. Our paper addresses property testing and parameter testing for permutations in a subpermutation perspective; more precisely, we investigate permutation properties and parameters that can be well-approximated based on randomly chosen subpermutations of much smaller size. In this context, we give a permutation counterpart of a famous result by Alon and Shapira [6] stating that every hereditary graph property is testable. Moreover, we develop a theory of convergence of permutation sequences, which is used to characterize testable permutation parameters along the lines of the work of Borgs et al. [12] in the case of graphs. This theory is interesting for its own sake, as it describes the closure of the set of all permutations as a special class of Lebesgue measurable functions in [0, 1]2, which in turn may be used to define a new model of random permutations.