The input in the Minimum-Cost Constraint Satisfaction Problem (MinCSP) over the Point Algebra contains a set of variables, a collection of constraints of the form $x < y$, $x = y$, $x \leq y$ and $x \neq y$, and a budget $k$. The goal is to check whether it is possible to assign rational values to the variables while breaking constraints of total cost at most $k$. This problem generalizes several prominent graph separation and transversal problems: MinCSP$(<)$ is equivalent to Directed Feedback Arc Set, MinCSP$(<,\leq)$ is equivalent to Directed Subset Feedback Arc Set, MinCSP$(=,\neq)$ is equivalent to Edge Multicut, and MinCSP$(\leq,\neq)$ is equivalent to Directed Symmetric Multicut. Apart from trivial cases, MinCSP$(\Gamma)$ for $\Gamma \subseteq \{<,=,\leq,\neq\}$ is NP-hard even to approximate within any constant factor under the Unique Games Conjecture. Hence, we study parameterized complexity of this problem under a natural parameterization by the solution cost $k$. We obtain a complete classification: if $\Gamma \subseteq \{<,=,\leq,\neq\}$ contains both $\leq$ and $\neq$, then MinCSP$(\Gamma)$ is W[1]-hard, otherwise it is fixed-parameter tractable. For the positive cases, we solve MinCSP$(<,=,\neq)$, generalizing the FPT results for Directed Feedback Arc Set and Edge Multicut as well as their weighted versions. Our algorithm works by reducing the problem into a Boolean MinCSP, which is in turn solved by flow augmentation. For the lower bounds, we prove that Directed Symmetric Multicut is W[1]-hard, solving an open problem.

翻译：最小代价约束满足问题（MinCSP）在点代数上的输入包含一组变量、形如$x < y$、$x = y$、$x \leq y$和$x \neq y$的约束集合，以及一个预算$k$。目标是判断是否可以对变量赋予有理数值，同时使违反约束的总代价不超过$k$。该问题推广了多个重要的图分割和横贯问题：MinCSP$(<)$等价于有向反馈弧集问题，MinCSP$(<,\leq)$等价于有向子集反馈弧集问题，MinCSP$(=,\neq)$等价于边多割问题，MinCSP$(\leq,\neq)$等价于有向对称多割问题。除平凡情况外，对于$\Gamma \subseteq \{<,=,\leq,\neq\}$，MinCSP$(\Gamma)$在唯一博弈猜想下即使近似到任意常数因子也是NP难的。因此，我们研究该问题在解代价$k$自然参数化下的参数化复杂度。我们获得了完整分类：若$\Gamma \subseteq \{<,=,\leq,\neq\}$同时包含$\leq$和$\neq$，则MinCSP$(\Gamma)$是W[1]-难的，否则是固定参数可解的。对于正例情形，我们解决了MinCSP$(<,=,\neq)$，推广了有向反馈弧集问题和边多割问题及其加权版本的FPT结果。我们的算法通过将问题归约为布尔MinCSP来工作，后者再通过流增广技术求解。对于下界，我们证明了有向对称多割问题是W[1]-难的，解决了一个开放问题。