Parameterized local search combines classic local search heuristics with the paradigm of parameterized algorithmics. While most local search algorithms aim to improve given solutions by performing one single operation on a given solution, the parameterized approach aims to improve a solution by performing $k$ simultaneous operations. Herein, $k$ is a parameter called search radius for which the value can be chosen by a user. One major goal in the field of parameterized local search is to outline the trade-off between the size of $k$ and the running time of the local search step. In this work, we introduce an abstract framework that generalizes natural parameterized local search approaches for a large class of partitioning problems: Given $n$ items that are partitioned into $b$ bins and a target function that evaluates the quality of the current partition, one asks whether it is possible to improve the solution by removing up to $k$ items from their current bins and reassigning them to other bins. Among others, our framework applies for the local search versions of problems like Cluster Editing, Vector Bin Packing, and Nash Social Welfare. Motivated by a real-world application of the problem Vector Bin Packing, we introduce a parameter called number of types $τ\le n$ and show that all problems fitting in our framework can be solved in $τ^k 2^{O(k)} |I|^{O(1)}$ time, where $|I|$ denotes the total input size. In case of Cluster Editing, the parameter $τ$ generalizes the well-known parameter neighborhood diversity of the input graph. We complement this by showing that for all considered problems, an algorithm significantly improving over our algorithm with running time $τ^k 2^{O(k)} |I|^{O(1)}$ would contradict the ETH. Additionally, we show that even on very restricted instances, all considered problems are W[1]-hard when parameterized by the search radius $k$ alone.

翻译：参数化局部搜索将经典局部搜索启发式与参数化算法范式相结合。传统局部搜索算法通常通过对给定解执行单一操作来改进解，而参数化方法则通过执行$k$个同步操作来优化解。其中$k$称为搜索半径参数，其值可由用户选择。该领域的主要目标之一是刻画参数$k$的规模与局部搜索步骤运行时间之间的权衡关系。本文提出一个抽象框架，将自然参数化局部搜索方法推广至一大类划分问题：给定划分为$b$个容器的$n$个物品，及评估当前划分质量的目标函数，问题为能否通过从当前容器中移除至多$k$个物品并重新分配至其他容器来改进解。该框架适用于聚类编辑、向量装箱和内什社会福利等问题的局部搜索变体。受向量装箱问题实际应用的启发，我们引入称为类型数$τ\le n$的参数，并证明所有符合框架的问题均可通过$τ^k 2^{O(k)} |I|^{O(1)}$时间求解，其中$|I|$表示总输入规模。在聚类编辑问题中，参数$τ$推广了众所周知的输入图邻域多样性参数。我们进一步证明：对于所有考虑的问题，若存在算法能在$τ^k 2^{O(k)} |I|^{O(1)}$时间上取得显著改进，则将与指数时间假设（ETH）相矛盾。此外，即使限制在极端简单实例上，所有考虑的问题在以搜索半径$k$为单一参数时均属于W[1]-难问题。