Constraint satisfaction problems form a nicely behaved class of problems that lends itself to complexity classification results. From the point of view of parameterized complexity, a natural task is to classify the parameterized complexity of MinCSP problems parameterized by the number of unsatisfied constraints. In other words, we ask whether we can delete at most $k$ constraints, where $k$ is the parameter, to get a satisfiable instance. In this work, we take a step towards classifying the parameterized complexity for an important infinite-domain CSP: Allen's interval algebra (IA). This CSP has closed intervals with rational endpoints as domain values and employs a set $A$ of 13 basic comparison relations such as ``precedes'' or ``during'' for relating intervals. IA is a highly influential and well-studied formalism within AI and qualitative reasoning that has numerous applications in, for instance, planning, natural language processing and molecular biology. We provide an FPT vs. W[1]-hard dichotomy for MinCSP$(\Gamma)$ for all $\Gamma \subseteq A$. IA is sometimes extended with unions of the relations in $A$ or first-order definable relations over $A$, but extending our results to these cases would require first solving the parameterized complexity of Directed Symmetric Multicut, which is a notorious open problem. Already in this limited setting, we uncover connections to new variants of graph cut and separation problems. This includes hardness proofs for simultaneous cuts or feedback arc set problems in directed graphs, as well as new tractable cases with algorithms based on the recently introduced flow augmentation technique. Given the intractability of MinCSP$(A)$ in general, we then consider (parameterized) approximation algorithms and present a factor-$2$ fpt-approximation algorithm.

翻译：约束满足问题构成了一类行为良好、易于进行复杂度分类结果的问题。从参数化复杂度的视角来看，一个自然的任务是分类以不满足约束数量为参数的MinCSP问题的参数化复杂度。换言之，我们询问是否能够删除最多$k$个约束（其中$k$是参数），从而得到一个可满足的实例。在本工作中，我们朝着为重要的无穷域CSP——Allen区间代数（IA）——分类其参数化复杂度迈出了一步。该CSP以具有有理数端点的闭区间作为域值，并使用一组包含13个基本比较关系的集合$A$（例如“先于”或“期间”）来关联区间。IA是人工智能和定性推理领域内一项极具影响力且被广泛研究的形式化方法，在规划、自然语言处理及分子生物学等多个方面有众多应用。我们为所有$\Gamma \subseteq A$下的MinCSP$(\Gamma)$问题提供了一个FPT与W[1]-难之间的二分判定。IA有时会扩展$A$中关系的并集或$A$上的一阶可定义关系，但要将我们的结果扩展到这些情况，需要先解决有向对称多割问题的参数化复杂度——这是一个著名的未解决问题。即便在此受限设定中，我们也发现了与图割和分离问题的新变种之间的联系，包括对有向图中同时割或反馈弧集问题的困难性证明，以及基于最近引入的流增强技术的新可解情况。鉴于MinCSP$(A)$在一般情况下难以处理，我们进一步考虑了（参数化）近似算法，并提出了一个因子为$2$的fpt-近似算法。