ABSTRACT
This paper studies approximation algorithms for problems on degree-bounded graphs. Let n and d be the number of vertices and the degree bound, respectively. This paper presents an algorithm to approximate the size of some maximal independent set with additive error ε n whose running time is O(d2). Using this algorithm, it also shows that there are approximation algorithms for many other problems, e.g., the maximum matching problem, the minimum vertex cover problem, and the minimum set cover problem, that run exponentially faster than existing algorithms with respect to d and 1/ε. Its approximation algorithm for the maximum matching problem can be transformed to a testing algorithm for the property of having a perfect matching with two-sided error. On the contrary, it also shows that every one-sided error tester for the property requires at least Ω(n) queries.
- Jonathan Aronson, Martin Dyer, Alan Frieze, and Stephen Suen. Randomized greedy matching. ii. Random Struct. Algorithms, 6(1):55--73, 1995. Google ScholarCross Ref
- Armen S. Asratian, Tristan M. J. Denley, and Roland Haggkvist. Bipartite graphs and their applications. Cambridge University Press, New York, NY, USA, 1998. Google ScholarDigital Library
- A. Czumaj and C. Sohler. Sublinear-time algorithms. Bulletin of the EATCS, 89:23--47, 2006.Google Scholar
- Martin Dyer and Alan Frieze. Randomized greedy matching. Random Struct. Algorithms, 2(1):29--45, 1991.Google ScholarCross Ref
- Oded Goldreich, Shari Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. J. ACM, 45(4):653--750, 1998. Google ScholarDigital Library
- Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York, NY, USA, 1979. Google ScholarDigital Library
- Oded Goldreich. Combinatorial property testing -- a survey. In In: Randomization Methods in Algorithm Design, pages 45--60. American Mathematical Society, 1998.Google ScholarCross Ref
- Oded Goldreich and Dana Ron. Property testing in bounded degree graphs. In STOC '97: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pages 406--415. ACM, 1997. Google ScholarDigital Library
- Piotr Indyk. Sublinear time algorithms for metric space problems. In STOC '99: Proceedings of the 31st annual ACM symposium on Theory of computing, pages 428--434, New York, NY, USA, 1999. ACM. Google ScholarDigital Library
- R. Kumar and R. Rubinfeld. Sublinear time algorithms. SIGACT News, 34:57--67, 2003. Google ScholarDigital Library
- László Lovász. On the ratio of optimal integral and fractional covers. SIAM J. Discret. Math., 13:383--390, 1975.Google ScholarDigital Library
- Sharon Marko and Dana Ron. Distance approximation in bounded-degree and general sparse graphs. APPROX-RANDOM, pages 475--486, 2006. Google ScholarDigital Library
- Huy N. Nguyen and Krzysztof Onak. Constant-time approximation algorithms via local improvements. In FOCS '08: Proceedings of the the 49th Annual Symposium on Foundations of Computer Science, 2008. to appear. Google ScholarDigital Library
- Michal Parnas and Dana Ron. Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms. Theor. Comput. Sci., 381(1--3):183--196, 2007. Google ScholarDigital Library
- Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput., 25(2):252--271, 1996. Google ScholarDigital Library
Index Terms
- An improved constant-time approximation algorithm for maximum~matchings
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