We consider the MIN-r-LIN(R) problem: given a system S of length-r linear equations over a ring R, find a subset of equations Z of minimum cardinality such that S-Z is satisfiable. The problem is NP-hard and UGC-hard to approximate within any constant even when r=|R|=2, so we focus on parameterized approximability with solution size as the parameter. For a large class of infinite rings R called Euclidean domains, Dabrowski et al. [SODA-2023] obtained an FPT-algorithm for MIN-2-LIN(R) using an LP-based approach based on work by Wahlstr\"om [SODA-2017]. Here, we consider MIN-r-LIN(R) for finite commutative rings R, initiating a line of research with the ultimate goal of proving dichotomy theorems that separate problems that are FPT-approximable within a constant from those that are not. A major motivation is that our project is a promising step for more ambitious classification projects concerning finite-domain MinCSP and VCSP. Dabrowski et al.'s algorithm is limited to rings without zero divisors, which are only fields among finite commutative rings. Handling zero divisors seems to be an insurmountable obstacle for the LP-based approach. In response, we develop a constant-factor FPT-approximation algorithm for a large class of finite commutative rings, called Bergen rings, and thus prove approximability for chain rings, principal ideal rings, and Z_m for all m>1. We complement the algorithmic result with powerful lower bounds. For r>2, we show that the problem is not FPT-approximable within any constant (unless FPT=W[1]). We identify the class of non-Helly rings for which MIN-2-LIN(R) is not FPT-approximable. Under ETH, we also rule out (2-e)-approximation for every e>0 for non-lineal rings, which includes e.g. rings Z_{pq} where p and q are distinct primes. Towards closing the gaps between upper and lower bounds, we lay the foundation of a geometric approach for analysing rings.

翻译：我们考虑MIN-r-LIN(R)问题：给定环R上长度为r的线性方程组S，求最小基数方程组子集Z，使得S-Z可满足。该问题即使在r=|R|=2时也是NP难且UGC难常数近似，因此我们关注以解规模为参数的参数化近似性。对于称为欧几里得域的一大类无限环R，Dabrowski等人[SODA-2023]基于Wahlström[SODA-2017]的线性规划方法，为MIN-2-LIN(R)提出了FPT算法。本文研究有限交换环R上的MIN-r-LIN(R)问题，旨在开启一系列以证明二分定理为最终目标的研究，从而区分常数FPT可近似与不可近似的问题。重要动机在于，本项目是面向有限域MinCSP和VCSP更宏大分类研究的关键推进步骤。Dabrowski等人的算法局限于无零因子环（在有限交换环中仅为域）。处理零因子似乎是线性规划方法难以逾越的障碍。为此，我们针对称为Bergen环的一大类有限交换环，开发了常数因子FPT近似算法，从而证明了链环、主理想环及所有m>1的Z_m环的可近似性。我们通过强下界结果补充算法结论：对于r>2，证明该问题不存在常数FPT近似（除非FPT=W[1]）；针对非Helly环类，证明MIN-2-LIN(R)不存在FPT近似；在ETH假设下，进一步排除非线性环（例如不同素数p,q构成的环Z_{pq}）的(2-ε)近似性。为弥合上下界间的差距，我们建立了分析环结构的几何方法基础。