# Testing k-Monotonicity

@article{Canonne2016TestingK, title={Testing k-Monotonicity}, author={Cl{\'e}ment L. Canonne and Elena Grigorescu and Siyao Guo and Akash Kumar and Karl Wimmer}, journal={Electron. Colloquium Comput. Complex.}, year={2016}, volume={TR16} }

A Boolean $k$-monotone function defined over a finite poset domain ${\cal D}$ alternates between the values $0$ and $1$ at most $k$ times on any ascending chain in ${\cal D}$. Therefore, $k$-monotone functions are natural generalizations of the classical monotone functions, which are the $1$-monotone functions. Motivated by the recent interest in $k$-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the…

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## References

SHOWING 1-10 OF 57 REFERENCES

### Sensitivity Conjecture and Log-Rank Conjecture for Functions with Small Alternating Numbers

- MathematicsICALP
- 2017

These results are extended to functions which alternate values for a relatively small number of times on any monotone path from 0-n to 1-n, and contribute to the recent line of research on functions with small alternating numbers.

### Monotonicity testing over general poset domains

- Mathematics, Computer ScienceSTOC '02
- 2002

It is shown 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.

### Transitive-Closure Spanners

- Computer Science, MathematicsSIAM J. Comput.
- 2012

The common task implicitly tackled in these diverse applications as the problem of constructing sparse TC-spanners is abstracted asThe study of approximability of the size of the sparsest of a given directed graph is initiated.

### Learning monotone decision trees in polynomial time

- Computer Science, Mathematics21st Annual IEEE Conference on Computational Complexity (CCC'06)
- 2006

This is the first algorithm that can learn arbitrary monotone Boolean functions to high accuracy, using random examples only, in time polynomial in a reasonable measure of the complexity of f.

### Monotonicity testing and shortest-path routing on the cube

- Mathematics, Computer ScienceComb.
- 2010

It is shown that for any δ > 0, the n-dimensional hypercube is not n-realizable with shortest paths, while previously it was only known that hypercubes are not 1- realizable with longest paths.

### New Algorithms and Lower Bounds for Monotonicity Testing

- Computer Science, Mathematics2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

A new lower bound is proved on the query complexity of any non-adaptive two-sided error algorithm for testing whether an unknown Boolean function f is monotone versus constant-far from monotones and an algorithm is presented that makes O(n<sup>7/8</sup>) poly(1/ε) queries.

### Learning circuits with few negations

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2014

This paper studies the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions.

### On learning monotone DNF under product distributions

- Mathematics, Computer ScienceInf. Comput.
- 2001

We show that the class of monotone 2O(√ n)-term DNF formulae can be PAC learned in polynomial time under the uniform distribution from random examples only. This is an exponential improvement over…

### Estimating the distance to a monotone function

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 2007

The running time of the algorithm is O(εf−1 log log εf− 1 log n), which is optimal within a factor of loglog ε f−1 and represents a substantial improvement over previous work.

### Correlation Bounds Against Monotone NC^1

- Mathematics, Computer ScienceComputational Complexity Conference
- 2015

The main theorem, proved using the pathset complexity framework introduced in [56], shows that the average-case k-CYCLE problem (on Erdos-Renyi random graphs with an appropriate edge density) is [EQUATION] hard for mNC1.