Almost Consistent Systems of Linear Equations

TL;DR

This paper investigates the parameterized complexity of the problem of minimizing unsatisfied equations in systems of linear equations, showing fixed-parameter tractability for equations with at most two variables over Euclidean domains, and W[1]-hardness otherwise.

Contribution

It introduces a fixed-parameter tractability result for two-variable equations over Euclidean domains and establishes W[1]-hardness for three or more variables, expanding understanding of this problem's complexity.

Findings

FPT algorithm for two-variable equations over Euclidean domains

W[1]-hardness for three or more variables

Generalization of important separators to biased graphs

Abstract

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.