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# NP-Hardness of Circuit Minimization for Multi-Output Functions

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## Acknowledgements

Igor C. Oliveira would like to thank Ján Pich and Rahul Santhanam for discussions on the complexity of circuit minimization for partial Boolean functions. Bruno Loff would like to thank Eric Allender for posing a question that inspired some of the results in this work, and the Higher School of Economics for inviting him to the conference "Randomness, Information, Complexity", in honor of Alexander Shen and Nikolay Vereshchagin’s 60th birthday, where said question was asked. Rahul Ilango would like to thank Eric Allender, Marco Carmosino, Russell Impagliazzo, Michael Saks, Rahul Santhanam, and Ryan Williams for their encouragement, suggestions, and helpful discussions.

## Cite As

Rahul Ilango, Bruno Loff, and Igor C. Oliveira. NP-Hardness of Circuit Minimization for Multi-Output Functions. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 22:1-22:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.CCC.2020.22

## Abstract

Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive. In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n → {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators. Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless 𝖯 = 𝖭𝖯, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational complexity and cryptography
##### Keywords
• MCSP
• circuit minimization
• communication complexity
• Boolean circuit

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