Checking whether a system of linear equations is consistent is a basic computational problem with ubiquitous applications. When dealing with inconsistent systems, one may seek an assignment that minimizes the number of unsatisfied equations. This problem is NP-hard and UGC-hard to approximate within any constant even for two-variable equations over the two-element field. We study this problem from the point of view of parameterized complexity, with the parameter being the number of unsatisfied equations. We consider equations defined over Euclidean domains - a family of commutative rings that generalize finite and infinite fields including the rationals, the ring of integers, and many other structures. We show that if every equation contains at most two variables, the problem is fixed-parameter tractable. This generalizes many eminent graph separation problems such as Bipartization, Multiway Cut and Multicut parameterized by the size of the cutset. To complement this, we show that the problem is W[1]-hard when three or more variables are allowed in an equation, as well as for many commutative rings that are not Euclidean domains. On the technical side, we introduce the notion of important balanced subgraphs, generalizing important separators of Marx [Theor. Comput. Sci. 2006] to the setting of biased graphs. Furthermore, we use recent results on parameterized MinCSP [Kim et al., SODA 2021] to efficiently solve a generalization of Multicut with disjunctive cut requests.
翻译:检查线性方程式系统是否一致是全局应用的基本计算问题。 在处理不一致的系统时, 人们可能会寻找一个能够将不满意方程式数量最小化的通货圈组合。 这个问题是NP- 硬和 UGC- 硬, 可以在任何常数中估计, 即使对两个元素字段的两种可变方程式也是如此。 我们从参数化复杂度的角度来研究这个问题, 参数是未满足方程式的数量 。 我们认为, 在 Euclidean 域上定义的等式是定义的。 在处理不一致的系统时, 人们可能会寻找一个能够将有限和无限的域( 包括理性、 整数环和许多其他结构) 的通货体组合。 我们显示, 如果每个等方程式都包含两个变量, 问题就是一个固定方程式。 我们从许多突出的图形分离问题, 如双向、 多路断线和多开的参数参数, 补充这个参数, 当三个或三个以上的变方程式被允许时, 我们发现问题是W[ 1- hard- hardalizal, commacial orizal orizal orizal orizal orizal eral eral 。 在 将许多 Cal- 将一个重要的 Sal- 将一个重要的 Scal- 的平流的平流图不是重要的平流的平流的平序 。