Abstract
For graphs G and T, and a family of graphs \(\mathcal {F}\) let \(\mathrm {ex}(G,T,\mathcal {F})\) denote the maximum possible number of copies of T in an \(\mathcal {F}\)-free subgraph of G. We investigate the algorithmic aspects of calculating and estimating this function. We show that for every graph T, finite family \(\mathcal {F}\) and constant \(\epsilon >0\) there is a polynomial time algorithm that approximates \(\mathrm {ex}(G,T,\mathcal {F})\) for an input graph G on n vertices up to an additive error of \(\epsilon n^{v(T)}\). We also consider the possibility of a better approximation, proving several positive and negative results, and suggesting a conjecture on the exact relation between T and \(\mathcal {F}\) for which no significantly better approximation can be found in polynomial time unless \(P=NP\).
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Notes
Monotone graph properties are properties that are closed under edge and vertex deletion, for example being \(K_3\) free or being planar.
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Acknowledgements
We thank two anonymous referees for helpful comments, and thank Lior Gishboliner for improving and simplifying the proof of Theorem 1.7
Funding
The funding was provided by Israel Science Foundation (IL) (Grant No. 281/17) and German-Israeli Foundation for Scientific Research and Development (IL) (Grant No. G-1347-304.6/2016).
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Noga Alon: Research supported in part by NSF Grant DMS-1855464 and BSF Grant 2018267. Clara Shikhelman: Research supported in part by an ISF grant.
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Alon, N., Shikhelman, C. Additive Approximation of Generalized Turán Questions. Algorithmica 84, 464–481 (2022). https://doi.org/10.1007/s00453-021-00899-4
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DOI: https://doi.org/10.1007/s00453-021-00899-4