VC-dimension of subsets of Hamming graphs
Christopher Housholder, Layna Mangiapanello, and Steven Senger
TL;DR
This paper investigates the VC-dimension of subsets within Hamming graphs, providing elementary methods that yield tight bounds and improve upon existing results for various parameters.
Contribution
It introduces new elementary techniques to analyze the VC-dimension of Hamming graphs, achieving tight bounds and extending prior work.
Findings
Tight bounds for VC-dimension 2 and 3 in H(2,q)
Elementary methods match or improve previous results
Results are tight for multiple parameters
Abstract
Following recent work on the VC-dimension of subsets of various pseudorandom graphs, we study the VC-dimension of Hamming graphs, which have proved somewhat resistant to the standard techniques in the literature. Our methods are elementary, and agree with or improve upon previously known results. In particular, for we show tight bounds on the size of a subset of vertices to guarantee VC-dimension 2 or 3. We also prove an assortment of results for other parameters, with many of these being tight as well.
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Taxonomy
TopicsCoding theory and cryptography · Graph Labeling and Dimension Problems · Cryptography and Data Security