On the testability and repair of hereditary hypergraph properties
Abstract
Recent works of AlonShapira and RödlSchacht have demonstrated that every hereditary property of undirected graphs or hypergraphs is testable with onesided error; informally, this means that if a graph or hypergraph satisfies that property "locally" with sufficiently high probability, then it can be perturbed (or "repaired") into a graph or hypergraph which satisfies that property "globally". In this paper we make some refinements to these results, some of which may be surprising. In the positive direction, we strengthen the results to cover hereditary properties of multiple directed polychromatic graphs and hypergraphs. In the case of undirected graphs, we extend the result to continuous graphs on probability spaces, and show that the repair algorithm is "local" in the sense that it only depends on a bounded amount of data; in particular, the graph can be repaired in a time linear in the number of edges. We also show that local repairability also holds for monotone or partite hypergraph properties (this latter result is also implicitly in work of Ishigami). In the negative direction, we show that local repairability breaks down for directed graphs, or for undirected 3uniform hypergraphs. The reason for this contrast in behavior stems from (the limitations of) Ramsey theory.
 Publication:

arXiv eprints
 Pub Date:
 January 2008
 arXiv:
 arXiv:0801.2179
 Bibcode:
 2008arXiv0801.2179A
 Keywords:

 Mathematics  Combinatorics;
 05C65;
 68R10;
 60C05
 EPrint:
 84 pages, 3 figures, to appear, Random Structures and Algorithms. Referee recommendations incorporated (in particular, a longer introduction)