Oded Goldreich
Shari Goldwasser
Dana Ron
Skip Abstract Section
Abstract
In this paper, we consider the question of determining whether a function f has property P or is ε-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.
In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.
- ALON, N., DUKE, R. A., LEFMANN, H., RODL, V., AND YUSTER, R. 1994. The algorithmic aspects of the regularity lemma. J. Algorithms 16, 80-109. Google Scholar
Digital Library
- ALON, N., GOLDREICH, O., H#STAD, J., AND PERALTA, R. 1992. Simple constructions of almost k-wise independent random variables. J. Rand. Struct. Algorithms 33, 289-304.Google ScholarCross Ref
- ANGLUIN, D. 1978. On the complexity of minimum inference of regular sets. Inf. Cont. 39, 337-350.Google ScholarCross Ref
- ARORA, S., FRIESE, A., AND KAPLAN, U. 1996. A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. In Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 21-30. Google Scholar
- ARORA, S., KARGER, D., AND KARPINSKI, M. 1995. Polynomial time approximation schemes for dense instances of NP-hard problems. In Proceedings of the 27th Annual ACM Symposium on the Theory of Computing (Las Vegas, Nev., May 29-June 1). ACM, New York, pp. 284-293. Google Scholar
- ARORA, S., LUND, C., MOTWANI, R., SUDAN, M., AND SZEGEDY, M. 1992. Proof verification and intractability of approximation problems. In Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 14-23.Google Scholar
- ARORA, S., AND SAFRA, S. 1992. Probabilistic checkable proofs: A new characterization of NP. In Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 1-13.Google Scholar
- BABAI, L., FORTNOW, L., LEVIN, L. A., AND SZEGEDY, M. 1991a. Checking computations in polylogarithmic time. In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing (New Orleans, La., May 6-8). ACM, New York, pp. 21-31. Google Scholar
- BABAI, L., FORTNOW, L., AND LUND, C. 1991b. Non-deterministic exponential time has two-prover interactive protocols. Computat. Complex. 1, 1, 3-40.Google ScholarCross Ref
- BELLARE, M., COPPERSMITH, D., H#STAD, J., KIWI, M., AND SUDAN, M. 1995a. Linearity testing in characteristic two. In Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 432-441. Google Scholar
- BELLARE, M., GOLDREICH, O., AND SUDAN, M. 1995b. Free bits, PCPs, and nonapproximability-- Towards tight results. In Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 422-431. (Full version available from ECCC, http:// www. eccc.uni-trier, de/eccc/. Google Scholar
- BELLARE, M., GOLDWASSER, S., LUND, C., AND RUSSELL, A. 1993. Efficient probabilistically checkable proofs and applications to approximation. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing (San Diego, Calif., May 16-18). ACM, New York, pp. 294-304. (Erratum in Proceedings of the 26th Annual ACM Symposium on the Theory of Computing. ACM, New York, 1994, p. 820.) Google Scholar
- BELLARE, M., AND SUDAN, M. 1994. Improved non-approximability results. In Proceedings of the 26th Annual ACM Symposium on the Theory of Computing (Montrdal, Que., Canada, May 23-25). ACM, New York, pp. 184-193. Google Scholar
- BEN-DAVID, S. 1992. Can finite samples detect singularities of real-valued functions? In Proceedings of the 24th Annual ACM Symposium on the Theory of Computing (Victoria, B.C., Canada, May 4-6). ACM, New York, pp. 390-399. Google Scholar
- BLUM, A., AND RIVEST, R. 1989. Training a 3-node neural network is NP-complete. In Advances in Neural Information Processing Systems I. Morgan-Kaufmann, San Mateo, Calif., pp. 494-501. Google Scholar
- BLUM, M., LUBY, M., AND RUBINFELD, R. 1993. Self-testing/correcting with applications to numerical problems. J. Comput. Syst. Sci. 47, 549-595. Google ScholarDigital Library
- BLUMER, A., EHRENFEUCHT, A., HAUSSLER, D., AND WARMUTH, M. K. 1989. Learnability and Vapnik-Chervonenkis dimension. J. ACM 36, 4 (Oct.), 929-965. Google ScholarDigital Library
- CANETTI, R., FEIGE, U., GOLDREICH, O., AND NAOR, M. 1996. Adaptively secure multi-party computation. Tech Rep. TR-682. Laboratory of Computer Science, Massachusetts Institute of Technology, Cambridge, Mass. Extended abstract in Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (Philadelphia, Pa., May 22-24) ACM, New York, pp. 639-648. Google Scholar
- CHERNOFF, H. 1952. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23, 493-507.Google ScholarCross Ref
- COVER, T.M. 1973. On determining the rationality of the mean of a random variable.Ann. Stat. 1, 862-871.Google ScholarCross Ref
- DE LA VEGA, W.F. 1994. MAX-CUT has a randomized approximation scheme in dense graphs. Rand. Struct. and Algorithms. 8, 4, 187-198. Google Scholar
- EDWARDS, K. 1986. The complexity of colouring problems on dense graphs. Theoret. Comput. Sci. 43, 337-343. Google ScholarCross Ref
- ERGUN, F., KANNAN, S., KUMAR, S. R., RUBINFELD, R., AND VISWANTHAN, M. 1998. Spot-checkers. In Proceedings of the 30th Annual ACM Symposium on the Theory of Computing. ACM, New York, to appear. Google Scholar
- FEIGE, U., GOLDWASSER, S., LOVASZ, L., SAFRA, S., AND SZEGEDY, M. 1991. Approximating Clique is almost NP-complete. In Proceedings of the 32nd Annual IEEE Symposium on Foundation of Computer Science. IEEE, New York, pp. 2-12. Google ScholarDigital Library
- FRIEZE, A., AND KANAN, R. 1996. The regularity lemma and approximation schemes for dense problems. In Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 12-20. Google Scholar
- GEMMELL, P., LIPTON, R., RUBINFELD, R., SUDAN, M., AND WIGDERSON, A. 1991. Self-testing/ correcting for polynomials and for approximate functions. In Proceedings of the 23rd Annual ACM Symposium on the Theory of Computing (New Orleans, La., May 6-8). ACM, New York, pp. 32-42. Google Scholar
- GOLD, M. E. 1978. Complexity of automation identification from given data. Inf. Cont. 37, 302-320.Google ScholarCross Ref
- GOLDREICH, O. 1995. Foundations of Crytography--Fragments of a Book. Available from ECCC, http ://www. eccc.uni-trier, de/eccc/.Google Scholar
- GOLDREICH, 0., AND RON, D. 1997. Property testing in bounded degree graphs. In Proceedings of the 29th Annual ACM Symposium on the Theory of Computing (El Paso, Tex., May 4-6). ACM, New York, pp. 406-415. Google Scholar
- GOLDREICH, 0., AND RON, D. 1998. A sublinear bipartite tester for bounded degree graphs. In Proceedings of the 30th Annual ACM Symposium on the Theory of Computing. ACM, New York, to appear. Google Scholar
- GROTSCHEL, M., LOVASZ, L., AND SCHRIJVER, A. 1988. Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, New York.Google Scholar
- HAJNAL, P. 1991. An #(n4/3) lower bound on the randomized complexity of graph properties. Combinatorica 11, 2, 131-144.Google ScholarCross Ref
- H#STAD, J. 1996a. Testing of the long code and hardness for clique. In Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (Philadelphia, Pa., May 22-24). ACM, New York, pp. 11-19. Google Scholar
- H#STAD, J. 1996b. Clique is hard to approximate within n1-E. In Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 627-636. Google Scholar
- H#STAD, J. 1997. Some optimal inapproximability results. In Proceedings of the 29th Annual ACM Symposium on the Theory of Computing (El Paso, Tex., May 4-6). ACM, New York, pp. 1-10. Google Scholar
- HOCHBAUM, D. S., AND SHMOYS, D.B. 1987. Using dual approximation algorithms for scheduling problems: Theoretical and practical results. J. ACM 34, 1 (Jan.), 144-162. Google ScholarDigital Library
- HOCHBAUM, D. S., AND SHMOYS, D.B. 1988. A polynomial approximation scheme for machine scheduling on uniform processors: Using the dual approximation approach. SIAM J. Comput. 17, 3, 539-551. Google ScholarDigital Library
- HOEFFDING, W., AND WOLFOWITZ, J. 1958. Distinguishability of sets of distributions. Ann. Math. Stat. 29, 700-718.Google ScholarCross Ref
- KARGER, D. R., MOTWANI, R., AND SUDAN, M. 1994. Approximate graph coloring by semidefinite programming. In Proceedings of the 35th Annual IEEE Symposium on the Foundation of Computer Science. ACM, New York, pp. 2-13.Google Scholar
- KEARNS, M. J., MANSOUR, Y., RON, D., RUBINFELD, R., SCHAPIRE, R. E., AND SELLIE, L. 1994. On the learnability of discrete distributions. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing (Montrdal, Que., Canada, May 23-25). ACM, New York, pp. 273-282. Google Scholar
- KEARNS, M., AND RON, D. 1998. Testing problems with sub-learning sample complexity. In Proceedings of the llth Annual ACM Symposium on Computational Learning Theory. ACM, New York, to appear. Google Scholar
- KEARNS, M. J., SCHAPIRE, R. E., AND SELLIE, L.M. 1992. Toward efficient agnostic learning. In Proceedings of the 5th Annual ACM Workshop on Computational Learning Theory (Pittsburgh, Pa., July 27-29). ACM, New York, pp. 341-351. Google Scholar
- KING, V. 1991. An O(n5/4) lower bound on the randomized complexity of graph properties. Combinatorica 11, 1, 23-32.Google ScholarCross Ref
- Kiwi, M. 1996. Probabilistically Checkable Proofs and the Testing of Hadamard-like Codes. Ph.D. dissertation. Massachusetts Institute of Technology, Cambridge, Mass. Google Scholar
- KULKARM, S. R., AND ZEITOUNI, 0. 1993. On probably correct classification of concepts. In Proceedings of the 6th Annual ACM Conference on Computational Learning Theory (Santa Cruz, Ca., July 26-28). ACM, New York, pp. 111-116. Google Scholar
- LOVASZ, L., AND YOUNG, N. 1991. Lecture notes on evasiveness of graph properties. Tech. Rep. TR-317-91. Computer Science Department, Princeton Univ., Princeton, N.J.Google Scholar
- NAOR, J., AND NAOR, M. 1993. Small-bias probability spaces: Efficient constructions and applications. SIAM J. Comput. 22, 4, 838-856. Google ScholarDigital Library
- PETRANK, E. 1994. The hardness of approximations: Gap location. Computat. Complex. 4, 133-157. Google ScholarDigital Library
- PITT, L., AND VALIANT, L.G. 1988. Computational limitations on learning from examples. J. ACM 35, 4 (Oct.), 965-984. Google ScholarDigital Library
- PITT, L., AND WARMUTH, M.K. 1993. The minimum consistent DFA problem cannot be approximated within an polynomial. J. ACM 40, 1 (Jan.), 95-142. Google ScholarDigital Library
- ROSENBERG, A. L. 1973. On the time required to recognize properties of graphs: A problem. SIGACT News 5, 15-16. Google ScholarDigital Library
- RUBINFELD, R. 1994. Robust functional equations and their applications to program testing. In Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 288-299.Google Scholar
- RUBINFELD, R., AND SUDAN, M. 1996. Robust characterization of polynomials with applications to program testing. SIAM J. Comput. 25, 2, 252-271. Google ScholarDigital Library
- SCHWARTZ, J. T. 1980. Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27, 4, (Oct.), 701-717. Google ScholarDigital Library
- SZEMEP, EDI, E. 1978. Regular partitions of graphs. In Proceedings of the Colloquim International CNRS. pp. 399-401.Google Scholar
- TREVISAN, L. 1998. Recycling queries in PCPs in linearity tests. In Proceedings of the 30th Annual ACM Symposium on the Theory of Computing, ACM, New York, to appear. Google Scholar
- VALIANT, L.G. 1984. A theory of the learnable. Commun. ACM 27, 11 (Nov.), 1134-1142. Google ScholarDigital Library
- VAPNIK, g. N., AND CHERVOENKIS, A.Y. 1971. On the uniform convergence of relative frequencies of events to their probabilities. Theory Prob. Applic. 17, 2, 264-280.Google ScholarCross Ref
- YAMANISHI, K. 1995. Probably almost discriminative learning. Mach. Learn. 18, 23-50. Google ScholarDigital Library
- YAO, A. C.C. 1987. Lower bounds to randomized algorithms for graph properties. In Proceedings of the 28th Annual IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 393-400.Google Scholar
- ZEITOUNI, 0., AND KULKARM, S.R. 1995. A general classification rule for probability measures. Ann. Stat. 23, 1393-1407.Google ScholarCross Ref
Index Terms
- Property testing and its connection to learning and approximation
Recommendations
L(2,1)-labeling of dually chordal graphs and strongly orderable graphs
An L(2,1)-labeling of a graph G=(V,E) is a function f:V(G)->{0,1,2,...} such that |f(u)-f(v)|>=2 whenever uv@__ __E(G) and |f(u)-f(v)|>=1 whenever u and v are at distance two apart. The span of an L(2,1)-labeling f of G, denoted as SP"2(f,G), is the ...
Property Testing in Bounded Degree Graphs
We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when ...
Property testing and its connection to learning and approximation
FOCS '96: Proceedings of the 37th Annual Symposium on Foundations of Computer ScienceThe authors study the question of determining whether an unknown function has a particular property or is /spl epsiv/-far from any function with that property. A property testing algorithm is given a sample of the value of the function on instances ...
Comments