Exact complexity of the compressed subword problem

Establish the exact computational complexity of the compressed subword problem: given two straight-line program (SLP) compressed words u and v over a finite alphabet, decide whether u embeds as a scattered subword in v, closing the current gap between the PP lower bound and the PSPACE upper bound.

Background

The paper’s PSPACE-hardness for directedness of context-free languages is obtained via a reduction that generalizes the compressed subword problem. The authors emphasize that, although the compressed subword problem is known to be in PSPACE and PP-hard, its exact complexity has remained a long-standing open problem.

Progress on this problem would not only settle a central question in algorithms on compressed strings but also clarify the boundary between PP and PSPACE in this domain, with implications for related inclusion problems between compressed ideals discussed in the paper.