## Abstract

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=n1,j=n2} is called a Monge matrix if for every 0 ≤ i < i′ ≤ n_{1} and 0 ≤ j < j′ ≤ n_{2} we have v_{i,j} + v_{i′,j′} ≤ v_{i′,j′} + v_{i′,j}. If inequality holds in the opposite direction, then 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 P is given query access to an unknown function f and a distance parameter ε. The algorithm should accept f with high probability if it has the property P and reject it with high probability if more than an ε-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] → ℝ is convex (concave) has query complexity and running time of O ((log n)/ε). Our algorithm for testing whether an n_{1} × n_{2} matrix V is a Monge (inverse Monge) matrix has query complexity and running time of O ((log n_{1} · log n_{2})/ε).

Original language | English |
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Pages (from-to) | 1158-1184 |

Number of pages | 27 |

Journal | SIAM Journal on Computing |

Volume | 32 |

Issue number | 5 |

DOIs | |

State | Published - Aug 2003 |

## Keywords

- Approximation algorithms
- Convex functions
- Monge matrices
- Property testing
- Randomized algorithms