Extend convolution tree decompositions with functional dependencies for exact-weight queries

Extend convolution tree decompositions and their associated evaluation techniques to incorporate functional dependencies in order to evaluate the 4-path exact-weight problem in time O(n^3), thereby matching the conditional lower bound under the k-SUM conjecture.

Background

The paper observes that exact-weight subgraph problems can be expressed as convolution sum-product queries, but the presented algorithms do not always match known conditional lower bounds. For the 4-path exact-weight problem, the relevant functional dependencies are A,B→X, B,C→Y, and C,D→−X−Y, where the edge endpoints determine the corresponding edge weight.

Using these dependencies suggests that the bags in a suitable convolution tree decomposition could be treated as having size three, yielding an O(n3) evaluation algorithm that matches the lower bound established under the k-SUM conjecture. The paper explicitly leaves the development of the required techniques for future work.

References

This suggests that it may be possible to extend CTDs to evaluate this query in time $O(n3)$, matching the lower bound in, but this requires new techniques like handling functional dependencies. We leave this for future work.

— Convolution Sum-Product Queries  (2609.03672 - Deeds et al., 3 Sep 2026) in Section 6, Conclusion and Future Work