A theory of NP-completeness and ill-conditioning for approximate real computations

Gregorio Malajovich; Mike Shub

doi:10.1145/3321479

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arXiv · 1803.03600

A theory of NP-completeness and ill-conditioning for approximate real computations

Gregorio Malajovich·Mike Shub

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Abstract

We develop a complexity theory for approximate real computations. We first produce a theory for exact computations but with condition numbers. The input size depends on a condition number, which is not assumed known by the machine. The theory admits deterministic and nondeterministic polynomial time recognizable problems. We prove that P is not NP in this theory if and only if P is not NP in the BSS theory over the reals. Then we develop a theory with weak and strong approximate computations. This theory is intended to model actual numerical computations that are usually performed in floating point arithmetic. It admits classes P and NP and also an NP-complete problem. We relate the P vs NP question in this new theory to the classical P vs NP problem.

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BibTeX RIS

Gregorio Malajovich, Mike Shub. 2018-03-09. A theory of NP-completeness and ill-conditioning for approximate real computations. https://doi.org/10.1145/3321479

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