Monotonicity testing over general poset domains

The field of property testing studies algorithms that distinguish, using a small number of queries, between inputs which satisfy a given property, and those that are `far' from satisfying the property. Testing properties that are defined in terms of monotonicity has been extensively investigated, primarily in the context of the monotonicity of a sequence of integers, or the monotonicity of a function over the n-dimensional hypercube {1,&ldots;,m}n. These works resulted in monotonicity testers whose query complexity is at most polylogarithmic in the size of the domain.We show that in its most general setting, testing that Boolean functions are close to monotone is equivalent, with respect to the number of required queries, to several other testing problems in logic and graph theory. These problems include: testing that a Boolean assignment of variables is close to an assignment that satisfies a specific 2-CNF formula, testing that a set of vertices is close to one that is a vertex cover of a specific graph, and testing that a set of vertices is close to a clique.We then investigate the query complexity of monotonicity testing of both Boolean and integer functions over general partial orders. We give algorithms and lower bounds for the general problem, as well as for some interesting special cases. In proving a general lower bound, we construct graphs with combinatorial properties that may be of independent interest.

[1]  R. Durrett Probability: Theory and Examples , 1993 .

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

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

[4]  Artur Czumaj,et al.  Property Testing in Computational Geometry , 2000, ESA.

[5]  N. Alon,et al.  The Probabilistic Method, Second Edition , 2000 .

[6]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

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

[8]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[9]  R. P. Dilworth,et al.  A DECOMPOSITION THEOREM FOR PARTIALLY ORDERED SETS , 1950 .

[10]  Ronitt Rubinfeld,et al.  Spot-Checkers , 2000, J. Comput. Syst. Sci..

[11]  Jirí Sgall,et al.  Functions that have read-twice constant width branching programs are not necessarily testable , 2004, Random Struct. Algorithms.

[12]  Eldar Fischer,et al.  Testing of matrix properties , 2001, STOC '01.

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

[14]  M. Sudan,et al.  Robust Characterizations of Polynomials and Their Applications to Program Testing , 1993 .

[15]  Avi Wigderson,et al.  Simple analysis of graph tests for linearity and PCP , 2003, Random Struct. Algorithms.

[16]  R. Tarjan,et al.  A Separator Theorem for Planar Graphs , 1977 .

[17]  E. Fischer THE ART OF UNINFORMED DECISIONS: A PRIMER TO PROPERTY TESTING , 2004 .

[18]  Eldar Fischer On the strength of comparisons in property testing , 2004, Inf. Comput..

[19]  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).

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

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

[22]  Ronitt Rubinfeld,et al.  Fast Approximate PCPs for Multidimensional Bin-Packing Problems , 1999, RANDOM-APPROX.

[23]  Noga Alon,et al.  Regular Languages are Testable with a Constant Number of Queries , 2000, SIAM J. Comput..

[24]  F. Behrend On Sets of Integers Which Contain No Three Terms in Arithmetical Progression. , 1946, Proceedings of the National Academy of Sciences of the United States of America.