[Paper Review] Counting Subgraphs in Degenerate Graphs

Lior Gishboliner, Yevgeny Levanzov+1 more|arXiv (Cornell University)|Jan 1, 2020

Limits and Structures in Graph Theory46 references4 citations

TL;DR

This paper fully characterizes the graphs H for which counting copies in degenerate graphs can be done in linear time, proving that a recently proposed sufficient condition for tractability is also necessary. It resolves open problems on induced subgraph counting and homomorphism counting in degenerate graphs using novel hardness techniques.

ABSTRACT

We consider the problem of counting the number of copies of a fixed graph $H$ within an input graph $G$. This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input $G$ has bounded degeneracy. This is a rich family of graphs, containing all graphs without a fixed minor (e.g. planar graphs), as well as graphs generated by various random processes (e.g. preferential attachment graphs). We say that $H$ is easy if there is a linear-time algorithm for counting the number of copies of $H$ in an input $G$ of bounded degeneracy. A seminal result of Chiba and Nishizeki from '85 states that every $H$ on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all $H$ on 5 vertices, and further proved that for every $k > 5$ there is a $k$-vertex $H$ which is not easy. They left open the natural problem of characterizing all easy graphs $H$. Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph $H$ to be easy. Here we show that this sufficient condition is also necessary, thus fully answering the Bera--Pashanasangi--Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms. Our proofs rely on several novel approaches for proving hardness results in the context of subgraph-counting.

Motivation & Objective

• To fully characterize all graphs H for which counting copies in degenerate graphs can be done in linear time.
• To resolve the open problem posed by Bera, Pashanasangi, and Seshadhri on characterizing 'easy' graphs H.
• To extend the characterization to induced subgraph counting and homomorphism counting in degenerate graphs.
• To establish both sufficiency and necessity of a recently proposed framework for subgraph counting tractability.

Proposed method

• Leveraging Bressan's framework for subgraph counting in degenerate graphs to derive a sufficient condition for linear-time tractability.
• Proving that the sufficient condition from Bressan's framework is also necessary, thereby fully characterizing easy graphs H.
• Developing novel hardness techniques tailored to subgraph-counting problems in the degeneracy setting.
• Analyzing structural properties of graphs H that determine their tractability in bounded degeneracy graphs.
• Using combinatorial and graph-theoretic arguments to establish tight bounds on the complexity of counting subgraphs.
• Extending the analysis to induced subgraph counting and homomorphism counting, showing analogous characterizations.

Experimental results

Research questions

• RQ1 Which graphs H admit a linear-time algorithm for counting their copies in graphs of bounded degeneracy?
• RQ2 Is the sufficient condition for tractability proposed by Bressan both necessary and sufficient for linear-time subgraph counting in degenerate graphs?
• RQ3 Can the characterization of easy graphs H be extended to induced subgraph counting in degenerate graphs?
• RQ4 Can the same framework be applied to homomorphism counting in degenerate graphs?
• RQ5 What structural properties of H determine whether it is easy to count in degenerate graphs?

Key findings

• The sufficient condition for linear-time subgraph counting in degenerate graphs proposed by Bressan is also necessary, fully characterizing all easy graphs H.
• All graphs H on at most 5 vertices are easy to count in degenerate graphs, confirming and extending prior results.
• For every k > 5, there exists a k-vertex graph H that is not easy to count, confirming the threshold behavior of the problem.
• The characterization of easy graphs H extends to induced subgraph counting, providing a complete classification.
• The framework also yields a complete characterization for homomorphism counting in degenerate graphs, with analogous necessary and sufficient conditions.
• The paper establishes new hardness techniques that are specifically tailored to subgraph-counting problems in the degeneracy setting.