Property Testing: A Learning Theory Perspective
Foundations and Trends® in Machine Learning
Subclasses of solvable problems from classes of combinatorial optimization problems
Cybernetics and Systems Analysis
FSTTCS'04 Proceedings of the 24th international conference on Foundations of Software Technology and Theoretical Computer Science
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Convex and submodular functions play an important role in many applications, and in particular in combinatorial optimization. Here we study two special cases: convexity in one dimension and submodularity in two dimensions. The latter type of functions are equivalent to the well-known Monge matrices. A matrix $V = \{v_{i,j}\}_{i,j=0}^{i=n_1,j=n_2}$ is called a Monge matrix if for every $0 \leq i V is an inverse Monge matrix (supermodular function). Many problems, such as the traveling salesperson problem and various transportation problems, can be solved more efficiently if the input is a Monge matrix.In this work we present testing algorithms for the above properties. A testing algorithm for a predetermined property $\cal P$ is given query access to an unknown function f and a distance parameter $\epsilon$. The algorithm should accept f with high probability if it has the property $\cal P$ and reject it with high probability if more than an $\epsilon$-fraction of the function values should be modified so that f obtains the property. Our algorithm for testing whether a 1-dimensional function $f:[n] \rightarrow \mathbb{R}$ is convex (concave) has query complexity and running time of $O\left((\log n) /\epsilon\right)$. Our algorithm for testing whether an n1 × n2 matrix V is a Monge (inverse Monge) matrix has query complexity and running time of $O\left((\log n_1\cdot\log n_2) /\epsilon\right)$.