# The Complexity of Partial Function Extension for Coverage Functions

## TL;DR

This paper investigates the computational complexity of extending partial functions to coverage functions, proving NP-completeness, and explores approximate extension notions with bounds, impacting learning theory and function extension problems.

## Contribution

It establishes NP-completeness of partial coverage function extension and analyzes approximate extension notions with tight bounds.

## Key findings

- Partial function extension to coverage functions is NP-complete.
- Existence of polynomial-sized certificates for extendibility.
- Bounds for approximate extension under multiplicative and additive errors.

## Abstract

Coverage functions are an important subclass of submodular functions, finding applications in machine learning, game theory, social networks, and facility location. We study the complexity of partial function extension to coverage functions. That is, given a partial function consisting of a family of subsets of $[m]$ and a value at each point, does there exist a coverage function defined on all subsets of $[m]$ that extends this partial function? Partial function extension is previously studied for other function classes, including boolean functions and convex functions, and is useful in many fields, such as obtaining bounds on learning these function classes. We show that determining extendibility of a partial function to a coverage function is NP-complete, establishing in the process that there is a polynomial-sized certificate of extendibility. The hardness also gives us a lower bound for learning coverage functions. We then study two natural notions of approximate extension, to account for errors in the data set. The two notions correspond roughly to multiplicative point-wise approximation and additive $L_1$ approximation. We show upper and lower bounds for both notions of approximation. In the second case we obtain nearly tight bounds.