Fixed-parameter counting from fixed-parameter decision

Determine whether fixed-parameter tractability of deciding whether a homomorphism exists from a pattern structure A to a host structure B implies fixed-parameter tractability of counting all homomorphisms from A to B for arbitrary recursively enumerable classes of pattern hypergraphs.

Background

The paper establishes that, on recursively enumerable classes of pattern hypergraphs with bounded λ\lambda, fixed-parameter tractability of homomorphism decision, fixed-parameter tractability of exact homomorphism counting, and polynomial-time exact counting are equivalent under ETH. The unrestricted implication remains unresolved, so the bounded-λ\lambda result does not settle the general case.

References

It is open in general whether fixed-parameter tractability of deciding $A\to B$ implies fixed-parameter tractability of counting all homomorphisms from $A$ to $B.

— FPT=PTIME for Homomorphism Problems on Sparse-Incidence and Bounded-Independence Patterns  (2609.21840 - Lanzinger, 18 Sep 2026) in Section 1, paragraph 3) Exact counting on bounded-$\lambda$ classes

Whether exact counting can achieve the same exponent remains open Section~8.

— FPT=PTIME for Homomorphism Problems on Sparse-Incidence and Bounded-Independence Patterns  (2609.21840 - Lanzinger, 18 Sep 2026) in Section 5, paragraph beginning "The algorithmic gap appears in PANDA"