Approximate Modularity

A set function on a ground set of size n is approximately modular if it satisfies every modularity requirement to within an additive error, approximate modularity is the set analog of approximate linearity. In this paper we study how close, in additive error, can approximately modular functions be to truly modular functions. We first obtain a polynomial time algorithm that makes O(n 2 log n) queries to any approximately modular function to reconstruct a modular function that is O(√n)-close. We also show an almost matching lower bound: any algorithm world need super polynomially many queries to construct a modular function that is o(√(n/log n))-close. In a striking contrast to these near-tight computational reconstruction bounds, we then show that for any approximately modular function, there exists a modular function that is O(log n)-close.

Authors

• Flavio Chierichetti
• Abhimanyu Das
• Anirban Dasgupta 0001
• Ravi Kumar 0001

Keywords

• Approximation methods
• Yttrium
• Polynomials
• Approximation algorithms
• Additives
• Linearity
• Manganese
• Additional Error
• Polynomial-time Algorithm
• Modularity Function
• Random Variables
• Upper Bound
• Linear Function
• Proof Of Theorem
• Smallest Value
• Set Of Elements
• Function Approximation
• Banach Space
• Maximum Absolute Value
• Convex Approximation
• Multiset
• Convex Domain
• modularity
• duality
• probabilistic method