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DOI: 10.4230/LIPIcs.MFCS.2016.63
On the Exact Learnability of Graph Parameters: The Case of Partition Functions

Authors: Nadia Labai and Johann A. Makowsky

Published in: LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Abstract
We study the exact learnability of real valued graph parameters f which are known to be representable as partition functions which count the number of weighted homomorphisms into a graph H with vertex weights alpha and edge weights beta. M. Freedman, L. Lovasz and A. Schrijver have given a characterization of these graph parameters in terms of the k-connection matrices C(f,k) of f. Our model of learnability is based on D. Angluin's model of exact learning using membership and equivalence queries. Given such a graph parameter f, the learner can ask for the values of f for graphs of their choice, and they can formulate hypotheses in terms of the connection matrices C(f,k) of f. The teacher can accept the hypothesis as correct, or provide a counterexample consisting of a graph. Our main result shows that in this scenario, a very large class of partition functions, the rigid partition functions, can be learned in time polynomial in the size of H and the size of the largest counterexample in the Blum-Shub-Smale model of computation over the reals with unit cost.
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Nadia Labai and Johann A. Makowsky. On the Exact Learnability of Graph Parameters: The Case of Partition Functions. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 63:1-63:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{labai_et_al:LIPIcs.MFCS.2016.63,
  author =	{Labai, Nadia and Makowsky, Johann A.},
  title =	{{On the Exact Learnability of Graph Parameters: The Case of Partition Functions}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{63:1--63:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.63},
  URN =		{urn:nbn:de:0030-drops-64750},
  doi =		{10.4230/LIPIcs.MFCS.2016.63},
  annote =	{Keywords: exact learning, partition function, weighted homomorphism, connection matrices}
}
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