TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhys.14.6.173
TI  - Many bounded versions of undecidable problems are NP-hard
PY  - 2023/06/29
UR  - https://www.scipost.org/SciPostPhys.14.6.173
JF  - SciPost Physics
JA  - SciPost Phys.
VL  - 14
IS  - 6
SP  - 173
A1  - Klingler, Andreas
AU  - van der Eyden, Mirte
AU  - Stengele, Sebastian
AU  - Reinhart, Tobias
AU  - de las Cuevas, Gemma
AB  - Several physically inspired problems have been proven undecidable; examples are the spectral gap problem and the membership problem for quantum correlations. Most of these results rely on reductions from a handful of undecidable problems, such as the halting problem, the tiling problem, the Post correspondence problem or the matrix mortality problem. All these problems have a common property: they have an NP-hard bounded version. This work establishes a relation between undecidable unbounded problems and their bounded NP-hard versions. Specifically, we show that NP-hardness of a bounded version follows easily from the reduction of the unbounded problems. This leads to new and simpler proofs of the NP-hardness of bounded version of the Post correspondence problem, the matrix mortality problem, the positivity of matrix product operators, the reachability problem, the tiling problem, and the ground state energy problem. This work sheds light on the intractability of problems in theoretical physics and on the computational consequences of bounding a parameter.
ER  -

@Article{10.21468/SciPostPhys.14.6.173,
	title={{Many bounded versions of undecidable problems are NP-hard}},
	author={Andreas Klingler and Mirte van der Eyden and Sebastian Stengele and Tobias Reinhart and Gemma de las Cuevas},
	journal={SciPost Phys.},
	volume={14},
	pages={173},
	year={2023},
	publisher={SciPost},
	doi={10.21468/SciPostPhys.14.6.173},
	url={https://scipost.org/10.21468/SciPostPhys.14.6.173},
}