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Faster Subgraph Counting in Sparse Graphs

Author Marco Bressan

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Marco Bressan
  • Department of Computer Science, Sapienza University of Rome, Italy

Funding

Marco Bressan is supported in part by the ERC Starting Grant DMAP 680153, by a Google Focused Research Award "ALL4AI", and by the "Dipartimenti di Eccellenza 2018-2022" grant awarded to the Department of Computer Science of the Sapienza University of Rome.

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Marco Bressan. Faster Subgraph Counting in Sparse Graphs. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.IPEC.2019.6

Abstract

A fundamental graph problem asks to compute the number of induced copies of a k-node pattern graph H in an n-node graph G. The fastest algorithm to date is still the 35-years-old algorithm by Nešetřil and Poljak [Nešetřil and Poljak, 1985], with running time f(k) * O(n^{omega floor[k/3] + 2}) where omega <=2.373 is the matrix multiplication exponent. In this work we show that, if one takes into account the degeneracy d of G, then the picture becomes substantially richer and leads to faster algorithms when G is sufficiently sparse. More precisely, after introducing a novel notion of graph width, the DAG-treewidth, we prove what follows. If H has DAG-treewidth tau(H) and G has degeneracy d, then the induced copies of H in G can be counted in time f(d,k) * O~(n^{tau(H)}); and, under the Exponential Time Hypothesis, no algorithm can solve the problem in time f(d,k) * n^{o(tau(H)/ln tau(H))} for all H. This result characterises the complexity of counting subgraphs in a d-degenerate graph. Developing bounds on tau(H), then, we obtain natural generalisations of classic results and faster algorithms for sparse graphs. For example, when d=O(poly log(n)) we can count the induced copies of any H in time f(k) * O~(n^{floor[k/4] + 2}), beating the Nešetřil-Poljak algorithm by essentially a cubic factor in n.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • subgraph counting
  • tree decomposition
  • degeneracy

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