The 2-Attractor Problem is NP-Complete
Janosch Fuchs, Philip Whittington
TL;DR
This work resolves the long-standing open question of the computational complexity of the $2$-attractor problem by proving $\mathrm{NP}$-completeness, and it situates this result within a broader framework that links string attractors to classical combinatorial problems. It introduces the $k$-set attractor and the colorful edge cover problem to refine reductions and capture the necessary structure, then builds a chain of reductions from $(3,B2)$-SAT to the $2$-attractor via a $2$-set attractor and a $2$-substring graph representation. The authors also establish APX-hardness for $k \ge 2$ and provide explicit inapproximability bounds, illustrating how attractor problems sit near the boundary of tractability and how colorfulness can illuminate complexity gaps. Collectively, the results close the last gap for $2$-attractors and supply a versatile reduction toolkit that could inform future work on related compressibility measures and their approximability.
Abstract
A $k$-attractor is a combinatorial object unifying dictionary-based compression. It allows to compare the repetitiveness measures of different dictionary compressors such as Lempel-Ziv 77, the Burrows-Wheeler transform, straight line programs and macro schemes. For a string $T \in Σ^n$, the $k$-attractor is defined as a set of positions $Γ\subseteq [1,n]$, such that every distinct substring of length at most $k$ is covered by at least one of the selected positions. Thus, if a substring occurs multiple times in $T$, one position suffices to cover it. A 1-attractor is easily computed in linear time, while Kempa and Prezza [STOC 2018] have shown that for $k \geq 3$, it is NP-complete to compute the smallest $k$-attractor by a reduction from $k$-set cover. The main result of this paper answers the open question for the complexity of the 2-attractor problem, showing that the problem remains NP-complete. Kempa and Prezza's proof for $k \geq 3$ also reduces the 2-attractor problem to the 2-set cover problem, which is equivalent to edge cover, but that does not fully capture the complexity of the 2-attractor problem. For this reason, we extend edge cover by a color function on the edges, yielding the colorful edge cover problem. Any edge cover must then satisfy the additional constraint that each color is represented. This extension raises the complexity such that colorful edge cover becomes NP-complete while also more precisely modeling the 2-attractor problem. We obtain a reduction showing $k$-attractor to be NP-complete and APX-hard for any $k \geq 2$.