Testing the (s, t)-disconnectivity of graphs and digraphs

Yuichi Yoshida, Yusuke Kobayashi

Abstract: Property testing is concerned with constant-time algorithms for deciding whether a given object satisfies a predetermined property or is far from satisfying it. In this paper, we consider testing properties related to the connectivity of two vertices in sparse graphs.We present one-sided error testers for (s,t)<math><mrow is="true"><mo is="true">(</mo><mi is="true">s</mi><mo is="true">,</mo><mi is="true">t</mi><mo is="true">)</mo></mrow></math>-disconnectivity with query complexity 2O(1/ϵ)<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">O</mi><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">/</mo><mi is="true">ϵ</mi><mo is="true">)</mo></mrow></mrow></msup></math> for digraphs and O(1/ϵ2)<math><mi is="true">O</mi><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">/</mo><msup is="true"><mrow is="true"><mi is="true">ϵ</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">)</mo></mrow></math> for graphs, where ϵ<math><mi is="true">ϵ</mi></math> is an error parameter. Furthermore, we show that these algorithms are the best possible in view of query complexity, i.e., we give matching lower bounds for two-sided error testers for both cases.We also give a constant-time algorithm for testing the (s,t)<math><mrow is="true"><mo is="true">(</mo><mi is="true">s</mi><mo is="true">,</mo><mi is="true">t</mi><mo is="true">)</mo></mrow></math>-disconnectivity of a directed bounded-degree hypergraph, which can be used to test the satisfiability of Horn SAT.