On testing convexity and submodularity

Michal Parnas*, Dana Ron, Ronitt Rubinfeld

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

26 Scopus citations

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 = {vi,j}i,j=0i=n1,j=n2 is called a Monge matrix if for every 0 ≤ i < i′ ≤ n1 and 0 ≤ j < j′ ≤ n2 we have vi,j + vi′,j′ ≤ vi′,j′ + vi′,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 n1 × n2 matrix V is a Monge (inverse Monge) matrix has query complexity and running time of O ((log n1 · log n2)/ε).

Original languageEnglish
Pages (from-to)1158-1184
Number of pages27
JournalSIAM Journal on Computing
Volume32
Issue number5
DOIs
StatePublished - Aug 2003

Keywords

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

Fingerprint

Dive into the research topics of 'On testing convexity and submodularity'. Together they form a unique fingerprint.

Cite this