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Keywords:

  • sublinear-time algorithms;
  • property testing;
  • bounded-degree graphs;
  • one-sided versus two-sided error probability

ABSTRACT

We present sublinear-time (randomized) algorithms for finding simple cycles of length at least inline image and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being Ck-minor free (resp., free from having the corresponding tree-minor). In particular, if the graph is inline image-far from being cycle-free (i.e., a constant fraction of the edges must be deleted to make the graph cycle-free), then the algorithm finds a cycle of polylogarithmic length in time inline image, where N denotes the number of vertices. This time complexity is optimal up to polylogarithmic factors.

The foregoing results are the outcome of our study of the complexity of one-sided error property testing algorithms in the bounded-degree graphs model. For example, we show that cycle-freeness of N-vertex graphs can be tested with one-sided error within time complexity inline image, where denotes the proximity parameter. This matches the known inline image query lower bound for one-sided error cycle-freeness testing, and contrasts with the fact that any minor-free property admits a two-sided error tester of query complexity that only depends on . We show that the same upper bound holds for testing whether the input graph has a simple cycle of length at least k, for any inline image. On the other hand, for any fixed tree T, we show that T-minor freeness has a one-sided error tester of query complexity that only depends on the proximity parameter .

Our algorithm for finding cycles in bounded-degree graphs extends to general graphs, where distances are measured with respect to the actual number of edges. Such an extension is not possible with respect to finding tree-minors in inline image complexity. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 139–184, 2014