Relate graph widths to fine-grained complexity reductions

Establish whether there exists a general rule connecting adaptive, normal, linear, entropic, and submodular graph-width notions with fine-grained complexity reductions.

Background

The paper separates five graph-width notions but does not determine how these parameters correspond to fine-grained lower-bound techniques. Existing lower bounds use clique embeddings, which are more restrictive than normal polymatroids, while other reductions based on the 3SUM conjecture appear related to linear width.

The authors identify a general relationship between graph-width notions and fine-grained complexity reductions as an open direction, motivated by the possibility of predicting which width parameter governs the complexity of graph pattern matching.