Approximately counting and sampling small witnesses using a colourful decision oracle

Dell, H., Lapinskas, J. and Meeks, K. (2022) Approximately counting and sampling small witnesses using a colourful decision oracle. SIAM Journal on Computing, 51(4), pp. 849-899. (doi: 10.1137/19M130604X)

Abstract

In this paper, we design efficient algorithms to approximately count the number of edges of a given $k$-hypergraph, and to sample an approximately uniform random edge. The hypergraph is not given explicitly and can be accessed only through its colorful independence oracle: The colorful independence oracle returns yes or no depending on whether a given subset of the vertices contains an edge that is colorful with respect to a given vertex-coloring. Our results extend and/or strengthen recent results in the graph oracle literature due to Beame et al. [ACM Trans. Algorithms, 16 (2020), 52], Dell and Lapinskas [Proceedings of STOC, ACM, 2018, pp. 281--288], and Bhattacharya et al. [Proceedings of ISAAC, 2019]. Our results have consequences for approximate counting/sampling: We can turn certain kinds of decision algorithms into approximate counting/sampling algorithms without causing much overhead in the running time. We apply this approximate counting/sampling-to-decision reduction to key problems in fine-grained complexity (such as $k$-SUM, $k$-OV, and weighted $k$-Clique) and parameterized complexity (such as induced subgraphs of size $k$ or weight-$k$ solutions to constraint satisfaction problems).

• Item Type: Articles
• Additional Information: The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) ERC grant agreement 334828. The research was also supported by a Royal Society of Edinburgh Personal Research Fellowship, funded by the Scottish Government.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Meeks, Dr Kitty
• Authors: Dell, H., Lapinskas, J., and Meeks, K.
• College/School: College of Science and Engineering > School of Computing Science
• Journal Name: SIAM Journal on Computing
• Publisher: Society for Industrial and Applied Mathematics
• ISSN: 0097-5397
• ISSN (Online): 1095-7111
• Published Online: 05 July 2022
• Copyright Holders: Copyright © 2022 Holger Dell, John Lapinskas, and Kitty Meeks
• First Published: First published in SIAM Journal on Computing 51(4):849-899
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher