Liu, AHH and Mande, NS 
ORCID: 0000-0002-9520-7340
(2026)
Instance complexity of Boolean functions
Discrete Mathematics and Theoretical Computer Science, 28 (2).
ISSN 1462-7264, 1365-8050
Abstract
In the area of query complexity of Boolean functions, the most widely studied cost measure of an algorithm is the worst-case number of queries made by it on an input. Motivated by the most natural cost measure studied in online algorithms, the competitive ratio, we consider a different cost measure for query algorithms for Boolean functions that captures the ratio of the cost of the algorithm and the cost of an optimal algorithm that knows the input in advance. The cost of an algorithm is its largest cost over all inputs. Grossman, Komargodski and Naor [ITCS’20] introduced this measure for Boolean functions, and dubbed it instance complexity. Grossman et al. showed, among other results, that monotone Boolean functions with instance complexity 1 are precisely those that depend on no variables or one variable. We complement the above-mentioned result of Grossman et al. by completely characterizing the instance complexity of symmetric Boolean functions. As a corollary we conclude that the only symmetric Boolean functions with instance complexity 1 are the Parity function and its complement. We also study the instance complexity of some graph properties like Connectivity and k-clique containment. In all the Boolean functions we study above, and those studied by Grossman et al., the instance complexity turns out to be the ratio of query complexity to minimum certificate complexity. It is a natural question to ask if this is the correct bound for all Boolean functions. We show a negative answer in a very strong sense, by analyzing the instance complexity of the Greater-Than and Odd-Max-Bit functions. We show that the above-mentioned ratio is linear in the input size for both of these functions, while we exhibit algorithms for which the instance complexity is a constant.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | 4901 Applied Mathematics, 4904 Pure Mathematics, 49 Mathematical Sciences |
| Divisions: | Faculty of Science & Engineering Faculty of Science & Engineering > School of Computer Science & Informatics Faculty of Science & Engineering > School of Computer Science & Informatics > Algorithms and Computing Systems |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 27 Jul 2026 16:03 |
| Last Modified: | 06 Aug 2026 11:01 |
| DOI: | 10.46298/dmtcs.15917 |
| Open Access URL: | https://dmtcs.episciences.org/18512 |
| Related Websites: | |
| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3199521 |
| Disclaimer: | The University of Liverpool is not responsible for content contained on other websites from links within repository metadata. Please contact us if you notice anything that appears incorrect or inappropriate. |
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