On the robust hardness of Grobner basis computation

The computation of Grobner bases is an established hard problem. By contrast with many other problems, however, there has been little investigation of whether this hardness is robust. In this paper, we frame and present results on the problem of approximate computation of Grobner bases. We show that it is NP-hard to construct a Grobner basis of the ideal generated by a set of polynomials, even when the algorithm is allowed to discard a 1 - epsilon fraction of the generators, and likewise when the algorithm is allowed to discard variables (and the generators containing them). Our results show that computation of Grobner bases is robustly hard even for simple polynomial systems (e.g. maximum degree 2, with at most 3 variables per generator). We conclude by greatly strengthening results for the Strong c-Partial Grobner problem posed by De Loera et al. [10]. Our proofs also establish interesting connections between the robust hardness of Grobner bases and that of SAT variants and graph-coloring. (C) 2018 Elsevier B.V. All rights reserved.