Beating treewidth in subgraph counting/homomorphism algorithms

Ascertain whether there exist algorithms that, for every fixed pattern graph H of treewidth t, count homomorphisms (or solve the corresponding subgraph counting problems) into an n-vertex graph G in time n^{o(t)} (up to polylogarithmic factors). This seeks to resolve the general “beat treewidth” question by demonstrating an exponent strictly sublinear in the treewidth for these problems or proving that such improvements are impossible under standard complexity assumptions.

Background

The authors’ lower bounds for type (III) instances match the brute-force exponent up to a 1/log k factor. They explicitly note that closing this gap relates to the longstanding open problem of whether one can “beat treewidth,” as formulated by Marx (2010), for subgraph counting and homomorphism problems.

Resolving this question would clarify whether the 1/log k slack is inherent or can be eliminated, and would have broad implications across parameterized and fine-grained complexity for graph pattern counting.

References

Moreover, the factor $1/\log k$ is not an artifact of our proofs, but a consequence of the notoriously open problem of whether “you can beat treewidth”.

— Parameterised Holant Problems (2409.13579 - Aivasiliotis et al., 20 Sep 2024) in Introduction, after Main Theorem (Section 1.3 ‘Our Contributions’), discussion of lower bounds following Theorem 1