eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-12-13
14:1
14:21
10.4230/LIPIcs.IPEC.2023.14
article
Computing Complexity Measures of Degenerate Graphs
Drange, Pål Grønås
1
https://orcid.org/0000-0001-7228-6640
Greaves, Patrick
2
Muzi, Irene
2
https://orcid.org/0000-0003-2410-6523
Reidl, Felix
2
https://orcid.org/0000-0002-2354-3003
University of Bergen, Norway
Birkbeck, University of London, UK
We show that the VC-dimension of a graph can be computed in time n^{⌈log d+1⌉} d^{O(d)}, where d is the degeneracy of the input graph. The core idea of our algorithm is a data structure to efficiently query the number of vertices that see a specific subset of vertices inside of a (small) query set. The construction of this data structure takes time O(d2^dn), afterwards queries can be computed efficiently using fast Möbius inversion.
This data structure turns out to be useful for a range of tasks, especially for finding bipartite patterns in degenerate graphs, and we outline an efficient algorithm for counting the number of times specific patterns occur in a graph. The largest factor in the running time of this algorithm is O(n^c), where c is a parameter of the pattern we call its left covering number.
Concrete applications of this algorithm include counting the number of (non-induced) bicliques in linear time, the number of co-matchings in quadratic time, as well as a constant-factor approximation of the ladder index in linear time.
Finally, we supplement our theoretical results with several implementations and run experiments on more than 200 real-world datasets - the largest of which has 8 million edges - where we obtain interesting insights into the VC-dimension of real-world networks.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol285-ipec2023/LIPIcs.IPEC.2023.14/LIPIcs.IPEC.2023.14.pdf
vc-dimension
datastructure
degeneracy
enumerating