Towards a Parameterized Approximation Dichotomy of MinCSP for Linear Equations over Finite Commutative Rings

Abstract

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 Wahlström [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.

Figures (7)

• Figure 1: Graphs $B_e$ corresponding to equations $x = 1 \cdot y$, $x = 2 \cdot y$, $x = 3 \cdot y$ and $x = 4 \cdot y$.
• Figure 2: Illustration of the geometric approach for two monomial rings $R=\mathbb{F}[\mathsf{x}\xspace,\mathsf{y}\xspace]/I$. The shaded region contains exponents $(a,b)$ such that $\mathsf{x}\xspace^a\mathsf{y}\xspace^b=0$ in $R$. The dotted line marks its convex hull.
• Figure 3: Let $S$ be the instance of $\textsc{$2$-Lin($\mathbb{Z}_4$)}$ with variables $a$ , $b$, $c$, $d$, $u$, $r$ and equations $a=1$, $b=1$, $2a=c$, $c=2u$, $u=r$, $2b=d$, and $d=r$. The figure illustrate the class assignment graph $G=G_I(S)$ for the ideal $I=(1)$. Note that $I$ has only two non-zero equivalence classes, namely, $E=\{2\}$ and $O=\{1,3\}$. Every edge of $G$ is annotated by the equation that implies it. A minimum conformal $st$-cut is given by the two edges that correspond to the equation $u=r$ and corresponds to the class assignment $a=O$, $b=O$, $c=E$, $d=E$, $u=O$, and $r=E$. Note that $G$ has only one minimal conformal $st$-cut closest to $s$, namely $\{a_Oc_E, b_Od_E\}$. This $st$-cut corresponds to the class assignment $a=O$, $b=O$, and $c=d=u=r=0$. Therefore, the optimum solution for $S$ only removes the equation $u=r$, however, any solution that corresponds to a minimum conformal $st$-cut closest to $s$ has to remove the equations $2a=c$ and $2b=d$.
• Figure 4: System of equations obtained from a pair of edges $p = \{e_1, e_2\}$ where $e_i = \{u_i, v_i\}$ in the proof of Theorem \ref{['thm:incomparable-annihilators']}. Edges $e_1$ and $e_2$ are illustrated by dashed lines, while the equations are illustrated by solid lines with labels describing equations between connected variables.
• Figure 5: Illustration of $R=\mathbb{F}[\mathsf{x}\xspace,\mathsf{y}\xspace]/(\mathsf{x}\xspace^4,\mathsf{x}\xspace\mathsf{y}\xspace^2,\mathsf{y}\xspace^3)$

Theorems & Definitions (124)

• Theorem 1: Theorem 3.1.4 in Bini:Flamini:FCR
• Proposition 2
• proof
• Theorem 4: Theorem 6.1 in Ganske:McDonald:rmjm73
• Theorem 5
• Corollary 6
• Proposition 7: also follows from Theorem 2.1 in Marks:Mazurek:ijm2016
• proof
• Proposition 8
• proof