Finding cycles and trees in sublinear time.
We present sublinear-time (randomized) algorithms for finding simple cycles of length at least k {ge} 3 and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being C{sub k}-minor-free (resp., free from having the corresponding tree-minor). In particular, if the graph is far (i.e., {Omega}(1)-far) from being cycle-free, i.e. if one has to delete a constant fraction of edges to make it cycle-free, then the algorithm finds a cycle of polylogarithmic length in time {tilde O}({radical}N), 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 {tilde O}(poly(1/{epsilon}) {center_dot} {radical}N). This matches the known {Omega}({radical}N) query lower bound, and contrasts with the fact that any minor-free property admits a two-sided error tester of query complexity that only depends on the proximity parameter {epsilon}. For any constant k {ge} 3, we extend this result to testing whether the input graph has a simple cycle of length at least k. 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 {epsilon}. 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 o({radical}N) complexity.
- Research Organization:
- Sandia National Laboratories (SNL), Albuquerque, NM, and Livermore, CA (United States)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- AC04-94AL85000
- OSTI ID:
- 1030364
- Report Number(s):
- SAND2010-7914C; TRN: US201124%%149
- Resource Relation:
- Conference: Proposed for presentation at the STOC 2011 Conference held June 6-8, 2011 in San Jose, CA.
- Country of Publication:
- United States
- Language:
- English
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