Testing properties of directed graphs: acyclicity and connectivity

This article initiates the study of testing properties of directed graphs. In particular, the article considers the most basic property of directed graphs—acyclicity. Because the choice of representation affects the choice of algorithm, the two main representations of graphs are studied. For the adjacency-matrix representation, most appropriate for dense graphs, a testing algorithm is developed that requires query and time complexity of Õ 1/ 2 , where is a distance parameter independent of the size of the graph. The algorithm, which can probe the adjacency matrix of the graph, accepts every graph that is acyclic, and rejects, with probability at least 2/3, every graph whose adjacency matrix should be modified in at least fraction of its entries so that it becomes acyclic. For the incidence list representation, most appropriate for sparse graphs, an V 1/3 lower bound is proved on the number of queries and the time required for testing, where V is the set of vertices in the graph. Along with acyclicity, this article considers the property of strong connectivity. Contrasting upper and lower bounds are proved for the incidence list representation. In particular, if the testing algorithm can query on both incoming and outgoing edges at each vertex, then it is possible to test strong connectivity in Õ 1/ time and query complexity. On the other hand, if the testing algorithm only has access to outgoing edges, then √N queries are required to test for strong connectivity. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20, 184–205, 2002 Correspondence to: Michael A. Bender; e-mail: bender@cs.sunysb.edu. ∗An earlier version of this article was presented at the 27th International Colloquium on Automata, Languages, and Programming (ICALP 2000). Contract grant sponsor: HRL Laboratories. Contract grant sponsor: ISX Corporation. Contract grant sponsor: Sandia National Laboratories. Contract grant sponsor: National Science Foundation. Contract grant number: EIA-0112849. Contract grant sponsor: Israel Science Foundation. Contract grant number: 32/00-1. © 2002 Wiley Periodicals, Inc. DOI 10.1002/rsa.10023

[1]  Dana Ron,et al.  Property Testing in Bounded Degree Graphs , 1997, STOC.

[2]  Dana Ron,et al.  Testing problems with sub-learning sample complexity , 1998, COLT' 98.

[3]  Joseph Naor,et al.  Approximating Minimum Feedback Sets and Multicuts in Directed Graphs , 1998, Algorithmica.

[4]  David B. Shmoys,et al.  A Polynomial Approximation Scheme for Machine Scheduling on Uniform Processors: Using the Dual Approximation Approach , 1986, FSTTCS.

[5]  Dana Ron,et al.  Improved Testing Algorithms for Monotonicity , 1999, Electron. Colloquium Comput. Complex..

[6]  David B. Shmoys,et al.  Using dual approximation algorithms for scheduling problems: Theoretical and practical results , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[7]  Refael Hassin,et al.  Approximations for the Maximum Acyclic Subgraph Problem , 1994, Inf. Process. Lett..

[8]  Bonnie Berger,et al.  Tight Bounds for the Maximum Acyclic Subgraph Problem , 1997, J. Algorithms.

[9]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[10]  David P. Williamson,et al.  Primal-Dual Approximation Algorithms for Feedback Problems in Planar Graphs , 1996, Comb..

[11]  Alan M. Frieze,et al.  Quick Approximation to Matrices and Applications , 1999, Comb..

[12]  Carsten Lund,et al.  Proof verification and the intractability of approximation problems , 1992, FOCS 1992.

[13]  Ronitt Rubinfeld,et al.  Spot-checkers , 1998, STOC '98.

[14]  Leonid A. Levin,et al.  Checking computations in polylogarithmic time , 1991, STOC '91.

[15]  Noga Alon,et al.  Regular languages are testable with a constant number of queries , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[16]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[17]  Dana Ron,et al.  A Sublinear Bipartiteness Tester for Bounded Degree Graphs , 1999, Comb..

[18]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[19]  Carsten Lund,et al.  Non-deterministic exponential time has two-prover interactive protocols , 2005, computational complexity.

[20]  Dana Ron,et al.  Testing the diameter of graphs , 1999, RANDOM-APPROX.

[21]  Yossi Matias,et al.  New sampling-based summary statistics for improving approximate query answers , 1998, SIGMOD '98.

[22]  Yossi Matias,et al.  DIMACS Series in Discrete Mathematicsand Theoretical Computer Science Synopsis Data Structures for Massive Data , 2007 .

[23]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1998, JACM.

[24]  Paul D. Seymour,et al.  Packing directed circuits fractionally , 1995, Comb..

[25]  Alan M. Frieze,et al.  A new rounding procedure for the assignment problem with applications to dense graph arrangement problems , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[26]  Ronitt Rubinfeld,et al.  Robust Characterizations of Polynomials with Applications to Program Testing , 1996, SIAM J. Comput..

[27]  Dana Ron,et al.  Testing Monotonicity , 2000, Comb..

[28]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[29]  Noga Alon,et al.  Efficient Testing of Large Graphs , 2000, Comb..