We study the parameterized complexity of MinCSP for so-called equality languages, i.e., for finite languages over an infinite domain such as $\mathbb{N}$, where the relations are defined via first-order formulas whose only predicate is $=$. This is an important class of languages that forms the starting point of all study of infinite-domain CSPs under the commonly used approach pioneered by Bodirsky, i.e., languages defined as reducts of finitely bounded homogeneous structures. Moreover, MinCSP over equality languages forms a natural class of optimisation problems in its own right, covering such problems as Edge Multicut, Steiner Multicut and (under singleton expansion) Edge Multiway Cut. We classify MinCSP$(\Gamma)$ for every finite equality language $\Gamma$, under the natural parameter, as either FPT, W[1]-hard but admitting a constant-factor FPT-approximation, or not admitting a constant-factor FPT-approximation unless FPT=W[2]. In particular, we describe an FPT case that slightly generalises Multicut, and show a constant-factor FPT-approximation for Disjunctive Multicut, the generalisation of Multicut where the ``cut requests'' come as disjunctions over $d = O(1)$ individual cut requests $s_i \neq t_i$. We also consider singleton expansions of equality languages, i.e., enriching an equality language with the capability for assignment constraints $(x=i)$ for either finitely or infinitely many constants $i \in \mathbb{N}$, and fully characterize the complexity of the resulting MinCSP.

翻译：我们研究了所谓等式语言（equality languages）的MinCSP参数复杂度，即定义在无穷域$\mathbb{N}$上的有限语言，其中所有关系通过仅含谓词$=$的一阶公式定义。这类语言是Bodirsky开创的无穷域CSP研究通用方法（即通过有限有界齐次结构的约简定义的语言）的重要起点。此外，等式语言上的MinCSP自身构成一类自然的优化问题，涵盖Edge Multicut、Steiner Multicut以及（通过单元素扩张的）Edge Multiway Cut等经典问题。我们针对每个有限等式语言$\Gamma$，在自然参数化下完整分类了MinCSP$(\Gamma)$的复杂度：要么属于FPT，要么是W[1]-难但允许常数因子FPT近似，要么不允许常数因子FPT近似（除非FPT=W[2]）。具体而言，我们描述了一个略推广Multicut的FPT案例，并针对析取型Multicut（即“割请求”表示为$d = O(1)$个个体割请求$s_i \neq t_i$的析取形式，是Multicut的推广）给出了常数因子FPT近似算法。我们还考虑了等式语言的单元素扩张，即通过为有限或无穷多个常数$i \in \mathbb{N}$添加赋值约束$(x=i)$来扩充等式语言，并完整刻画了由此产生的MinCSP的复杂度。