Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, October 06, 2020, 12:15 pm

Duration: 30 minutes

Location: Zoom: conference room

Speaker: Lior Gishboliner

Counting Subgraphs in Degenerate Graphs

We consider the problem of counting the number of copies of a fixed graph H in 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 forms a rich class of graphs, which includes for example all non-trivial minor-closed classes (and in particular planar graphs). Bera, Pashanasangi, and Seshadhri recently raised the question of characterizing the graphs H such that counting H-copies in a bounded-degeneracy input graph can be done in linear time. In this work we fully resolve this question, as well as the analogous questions for counting H-homomorphisms and induced H-copies. A recent result of Bressan shows that one can count H-homomorphisms in a bounded-degeneracy graph in time $O(n^{\tau(H)})$, where $\tau$ is a suitable width parameter, called DAG treewidth. In particular, H-homomorphisms can be counted in linear time if $\tau(H) = 1$. We show that this sufficient condition is also necessary, namely that H-homomorphisms can be counted in linear time if and only if $\tau(H) = 1$. We then use this to resolve the aforementioned problem of Bera, Pashanasangi, and Seshadhri. Our proofs rely on several new ideas for proving hardness results in the context of subgraph-counting. This is a joint work with Yevgeny Levanzov and Asaf Shapira.

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