Testing Eulerianity and connectivity in directed sparse graphs

Abstract

Property testing problems are relaxations of decision problems. A property testing algorithm (referred to as a testing algorithm or tester) has to decide if a given object has a prespecified property or is ϵ-far from the property (for a given distance parameter ϵ, and for a prespecified distance measure). The tester is given query access to the input, and is required to run in sublinear time.In this paper, we focus on testing properties of directed graphs (digraphs). In particular, we present the following results (where n is the number of vertices in the graph, d is the maximum degree, and davg is the average degree). •We present a testing algorithm for the property of Eulerianity in bounded-degree digraphs, which runs in time11The notation Õ(g(k)) stands for O(g(k))⋅polylog(g(k)).Õ(1/ϵ). For unbounded-degree digraphs, we show a lower bound of Ω(n/ϵ), and give a testing algorithm that runs in time Õ(n/ϵ3/2).•We study the property of k-vertex-connectivity, and give testing algorithms for both bounded-degree and unbounded-degree digraphs that run in time Õ((ckϵd)kd) and Õ((ckϵdavg)k+1), respectively (where c>1 is a constant). In addition, we give a simpler analysis of the testing algorithm for k-vertex-connectivity in bounded-degree undirected graphs that was shown by Yoshida and Ito [Y. Yoshida, H. Ito, Property testing on k-vertex-connectivity of graphs, in: ICALP’08: Proceedings of the 35th International Colloquium on Automata, Languages and Programming, Part I, Springer-Verlag, Berlin, Heidelberg, 2008, pp. 539–550] and extend the result to unbounded-degree undirected graphs.•We consider the property of k-edge-connectivity in digraphs, and simplify the analysis of the algorithm of Yoshida and Ito [Y. Yoshida, H. Ito, Testing k-edge-connectivity of digraphs, Journal of System Science and Complexity 23 (1) (2010) 91–101] for this property. In addition, we give a simpler analysis for the correctness of the testing algorithm for k-edge-connectivity in undirected graphs that was introduced by Goldreich and Ron [O. Goldreich, D. Ron, Property testing in bounded degree graphs, Algorithmica (2002) 302–343].