近似格罗布纳基计算的推广硬度结果
Extended Hardness Results for Approximate Gröbner Basis Computation
摘要：
最近提出了两个模型来探索格罗布纳基计算的鲁棒性。给定一个多项式系统，这两个模型都允许算法选择性地忽略一些多项式：算法只负责返回剩余多项式生成的理想的格罗布纳基。对于$q$ -分数格罗布纳基问题，允许算法忽略多项式的常数$(1-q)$ -分数（受一个自然结构约束）。在这里，我们证明了一个新的最强参数结果：即使允许算法选择要忽略的多项式的$(3/10-ε)$分数，并且仅需要相对于剩余多项式的某些词典顺序计算格罗布纳基，这也无法在多项式时间内完成（除非$P=NP$）。即使每个多项式的最大阶数为3，这个命题也成立。接下来，我们证明了最大次数为2的多项式系统的第一个鲁棒硬度结果：对于$q$ -分数模型，可以忽略多项式的$(1/5-ε)$分数而不丢失可证明的NP-硬度。这两个定理成立，即使每个多项式包含最多三个不同的变量。最后，对于De Loera等人的强$c$ -部分格罗布纳基问题，我们给出了依赖于Khot和Dinur等人的著名（未解决）解的条件结果.
英文摘要：
Two models were recently proposed to explore the robust hardness of Gröbner 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 Gröbner basis for the ideal generated by the remaining polynomials. For the $q$-Fractional Gröbner 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 $(3/10-ε)$-fraction of the polynomials to ignore, and need only compute a Gröbner 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 $(1/5-ε)$ 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 Gröbner Basis Problem of De Loera et al. we give conditional results that depend on famous (unresolved) conjectures of Khot and Dinur, et al.
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