2 research outputs found
Binary pattern tile set synthesis is NP-hard
In the field of algorithmic self-assembly, a long-standing unproven
conjecture has been that of the NP-hardness of binary pattern tile set
synthesis (2-PATS). The -PATS problem is that of designing a tile assembly
system with the smallest number of tile types which will self-assemble an input
pattern of colors. Of both theoretical and practical significance, -PATS
has been studied in a series of papers which have shown -PATS to be NP-hard
for , , and then . In this paper, we close the
fundamental conjecture that 2-PATS is NP-hard, concluding this line of study.
While most of our proof relies on standard mathematical proof techniques, one
crucial lemma makes use of a computer-assisted proof, which is a relatively
novel but increasingly utilized paradigm for deriving proofs for complex
mathematical problems. This tool is especially powerful for attacking
combinatorial problems, as exemplified by the proof of the four color theorem
by Appel and Haken (simplified later by Robertson, Sanders, Seymour, and
Thomas) or the recent important advance on the Erd\H{o}s discrepancy problem by
Konev and Lisitsa using computer programs. We utilize a massively parallel
algorithm and thus turn an otherwise intractable portion of our proof into a
program which requires approximately a year of computation time, bringing the
use of computer-assisted proofs to a new scale. We fully detail the algorithm
employed by our code, and make the code freely available online
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum