Improved Constant-Time Approximation Algorithms for Maximum Matchings and Other Optimization Problems
Abstract
We study constant-time approximation algorithms for bounded-degree graphs, which run in time independent of the number of vertices $n$. We present an algorithm that decides whether a vertex is contained in a some fixed maximal independent set with expected query complexity $O(d^2)$, where $d$ is the degree bound. Using this algorithm, we show constant-time approximation algorithms with certain multiplicative error and additive error $\epsilon n$ 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 $\frac{1}{\epsilon}$. Our approximation algorithm for the maximum matching problem can be transformed to a two-sided error tester for the property of having a perfect matching. On the contrary, we show that every one-sided error tester for the property requires at least $\Omega(n)$ queries.
1. , On constant time approximation of parameters of bounded degree graphs , Property Testing , 6390 ( 2011 ), pp. 234 -- 239 .
2. , Randomized greedy matching II , Random Structures Algorithms , 6 ( 1995 ), pp. 55 -- 73 .
3. , Bipartite Graphs and Their Applications , Cambridge University Press , Cambridge, UK , 1998 .
4. , Sublinear-time algorithms , Bull. Euro. Assoc. Theor. Comput. Sci. EATCS , 89 ( 2006 ), pp. 23 -- 47 .
5. , Randomized greedy matching , Random Structures Algorithms , 2 ( 1991 ), pp. 29 -- 45 .
6. , Computers and Intractability: A Guide to the Theory of NP-Completeness , W. H. Freeman , New York , 1979 .
7. ,
Combinatorial property testing - a survey , in Randomization Methods in Algorithm Design , American Mathematical Society , Providence, RI , 1998 , pp. 45 -- 60 .8. , Introduction to testing graph properties , Property Testing , 6390 ( 2011 ), pp. 105 -- 141 .
9. , Property testing and its connection to learning and approximation , J. ACM , 45 ( 1998 ), pp. 653 -- 750 .
10. , Property testing in bounded degree graphs , Algorithmica , 32 ( 2002 ), pp. 302 -- 343 .
11. , Sublinear time algorithms , SIGACT News , 34 ( 2003 ), pp. 57 -- 67 .
12. , On the ratio of optimal integral and fractional covers , SIAM J. Discrete Math. , 13 ( 1975 ), pp. 383 -- 390 .
13. , Approximating the distance to properties in bounded-degree and general sparse graphs , ACM Trans. Algorithms , 5 ( 2009 ), pp. 1 -- 28 .
14. , Constant-time approximation algorithms via local improvements , in
Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science , 2008 , pp. 327 -- 336 .15. , A near-optimal sublinear-time algorithm for approximating the minimum vertex cover size , in
Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms , 2012 , pp. 1123 -- 1131 .16. , Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms , Theoret. Comput. Sci. , 381 ( 2007 ), pp. 183 -- 196 .
17. , Algorithmic and analysis techniques in property testing , Found. Trends Theor. Comput. Sci. , 5 ( 2010 ), pp. 73 -- 205 .
18. , Robust characterizations of polynomials with applications to program testing , SIAM J. Comput. , 25 ( 1996 ), pp. 252 -- 271 .
