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# Partial Function Extension with Applications to Learning and Property Testing

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LIPIcs.ISAAC.2020.46.pdf
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## Cite As

Umang Bhaskar and Gunjan Kumar. Partial Function Extension with Applications to Learning and Property Testing. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ISAAC.2020.46

## Abstract

Partial function extension is a basic problem that underpins multiple research topics in optimization, including learning, property testing, and game theory. Here, 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 linearity. We formally study partial function extension to fundamental properties in combinatorial optimization - subadditivity, XOS, and matroid independence. A priori, it is not clear if partial function extension for these properties even lies in NP (or coNP). Our contributions are twofold. Firstly, for the properties studied, we give bounds on the complexity of partial function extension. For subadditivity and XOS, we give tight bounds on approximation guarantees as well. Secondly, we develop new connections between partial function extension and learning and property testing, and use these to give new results for these problems. In particular, for subadditive functions, we give improved lower bounds on learning, as well as the first subexponential-query tester.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Approximation algorithms analysis
##### Keywords
• Partial function extension
• matroid rank
• approximation algorithms
• learning
• property testing

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