In many real world networks, there already exists a (not necessarily optimal) $k$-partitioning of the network. Oftentimes, one aims to find a $k$-partitioning with a smaller cut value for such networks by moving only a few nodes across partitions. The number of nodes that can be moved across partitions is often a constraint forced by budgetary limitations. Motivated by such real-world applications, we introduce and study the $r$-move $k$-partitioning~problem, a natural variant of the Multiway cut problem. Given a graph, a set of $k$ terminals and an initial partitioning of the graph, the $r$-move $k$-partitioning~problem aims to find a $k$-partitioning with the minimum-weighted cut among all the $k$-partitionings that can be obtained by moving at most $r$ non-terminal nodes to partitions different from their initial ones. Our main result is a polynomial time $3(r+1)$ approximation algorithm for this problem. We further show that this problem is $W[1]$-hard, and give an FPTAS for when $r$ is a small constant.

翻译：在许多现实世界的网络中，已经存在一个（不一定最优的）$k$-划分。通常，我们希望通过仅移动少量节点跨越划分，来为这类网络找到具有更小割值的$k$-划分。可跨越划分移动的节点数量往往受预算限制的约束。受此类实际应用的启发，我们引入并研究了$r$-移动$k$-划分问题，这是多路割问题的一个自然变体。给定一个图、一组$k$个终端以及该图的初始划分，$r$-移动$k$-划分问题旨在在所有可通过将最多$r$个非终端节点移动到与其初始划分不同的划分而获得的$k$-划分中，找到具有最小加权割的$k$-划分。我们的主要结果是针对该问题的一个多项式时间$3(r+1)$近似算法。我们进一步证明该问题是$W[1]$-难的，并针对$r$为小常数的情况给出了一个全多项式时间近似方案（FPTAS）。