Extended Hardness Results for Approximate Gr\"obner Basis Computation

TL;DR

This paper establishes new hardness results for approximate Gr"obner basis computation, showing that even with significant polynomial ignoring, the problem remains computationally intractable under certain conditions.

Contribution

It proves the strongest known intractability results for approximate Gr"obner basis problems, including for degree-3 and degree-2 polynomial systems, with implications for related models.

Findings

Polynomial-time algorithms unlikely for 3/10-ignored degree-3 systems

NP-hardness persists for 1/5-ignored degree-2 systems

Conditional hardness results depend on unresolved conjectures

Abstract

Two models were recently proposed to explore the robust hardness of Gr\"obner basis computation. Given a polynomial system, both models allow an algorithm to selectively ignore some of the polynomials: the algorithm is only responsible for returning a Gr\"obner basis for the ideal generated by the remaining polynomials. For the $q$-Fractional Gr\"obner Basis Problem the algorithm is allowed to ignore a constant $(1-q)$-fraction of the polynomials (subject to one natural structural constraint). Here we prove a new strongest-parameter result: even if the algorithm is allowed to choose a $(3/10-\epsilon)$-fraction of the polynomials to ignore, and need only compute a Gr\"obner basis with respect to some lexicographic order for the remaining polynomials, this cannot be accomplished in polynomial time (unless $P=NP$). This statement holds even if every polynomial has maximum degree 3. Next, we prove the first robust hardness result for polynomial systems of maximum degree 2: for the $q$-Fractional model a $(1/5-\epsilon)$ fraction of the polynomials may be ignored without losing provable NP-Hardness. Both theorems hold even if every polynomial contains at most three distinct variables. Finally, for the Strong $c$-partial Gr\"obner Basis Problem of De Loera et al. we give conditional results that depend on famous (unresolved) conjectures of Khot and Dinur, et al.