ON WEIGHTED GRAPH SEPARATION PROBLEMS AND FLOW AUGMENTATION

One of the first applications of the recently introduced technique of flow augmentation [Kim et al., STOC 2022] is a fixed-parameter algorithm for the weighted version of DIRECTED FEEDBACK VERTEX SET, a landmark problem in parameterized complexity. In this article, we explore the applicability of flow augmentation to other weighted graph separation problems parameterized by the size of the cutset. We show the following: \bullet In weighted undirected graphs, MULTICUT is fixed-parameter tractable (FPT) in both the edge- and the vertex-deletion version. \bullet The weighted version of GROUP FEEDBACK VERTEX SET is FPT, even with oracle access to group operations. \bullet The weighted version of DIRECTED SUBSET FEEDBACK VERTEX SET is FPT. Our study reveals DIRECTED SYMMETRIC MULTICUT as the next important graph separation problem whose parameterized complexity remains unknown, even in the unweighted setting.