# Partial Function Extension with Applications to Learning and Property Testing

## TL;DR

This paper investigates the problem of extending partial functions to satisfy properties like subadditivity, submodularity, and convexity, providing complexity bounds, algorithms, and applications in learning and property testing.

## Contribution

It offers new complexity results, algorithms, and testing methods for extending partial functions with key properties, advancing understanding in optimization and learning.

## Key findings

- Extension for subadditive functions is coNP-complete with tight approximability bounds.
- Algorithms for submodular extension are effective in specific cases, but the general complexity remains open.
- Determining the existence of convex function extensions is efficient, but computing their values is NP-hard.

## Abstract

In partial function extension, we are given a partial function consisting of $n$ points from a domain and a function value at each point. Our objective is to determine if this partial function can be extended to a function defined on the domain, that additionally satisfies a given property, such as convexity. This basic problem underlies research questions in many areas, such as learning, property testing, and game theory. We formally study the problem of extending partial functions to satisfy fundamental properties in combinatorial optimization, focusing on upper and lower bounds for extension and applications to learning and property testing. (1) For subadditive functions, we show the extension problem is coNP-complete, and we give tight bounds on the approximability. We also give an improved lower bound for learning subadditive functions, and give the first nontrivial testers for subadditive and XOS functions. (2) For submodular functions, we show that if a partial function can be extended to a submodular function on the lattice closure (the minimal set that contains the partial function and is closed under union and intersection) of the partial function, it can be extended to a submodular function on the entire domain. We obtain algorithms for determining extendibility in a number of cases, including if $n$ is a constant, or the points are nearly the same size. The complexity of extendibility is in general unresolved. (3) Lastly, for convex functions in $\mathbb{R}^m$, we show an interesting juxtaposition: while we can determine the existence of an extension efficiently, computing the value of a widely-studied convex extension at a given point is strongly NP-hard.