2025/05/27 by Nadym Mallek, Mallek, Nadym, Kirill Simonov +1
Computer Science · Engineering · #Advanced Graph Theory Research #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #VLSI and FPGA Design Techniques
paper · pdf · doi:10.48550/arxiv.2505.21433
openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the Requirement Cut problem, a generalization of numerous classical graph partitioning problems including Multicut, Multiway Cut, k-Cut, and Steiner Multicut among others. Given a graph with edge costs, terminal groups (S1, ..., Sg) and integer requirements (r1,... , rg); the goal is to compute a minimum-cost edge cut that separates each group Si into at least ri connected components. Despite many efforts, the best known approximation for Requirement Cut yields a double-logarithmic O(log(g).log(n)) approximation ratio as it relies on embedding general graphs into trees and solving the tree instance. In this paper, we explore two largely unstudied structural parameters in order to obtain single-logarithmic approximation ratios: (1) the number of minimal Steiner trees in the instance, which in particular is upper-bounded by the number of spanning trees of the graphs multiplied by g, and (2) the depth of series-parallel graphs. Specifically, we show that if the number of minimal Steiner trees is polynomial in n, then a simple LP-rounding algorithm yields an O(log n)-approximation, and if the graph is series-parallel with a constant depth then a refined analysis of a known probabilistic embedding yields a O(depth.log(g))-approximation on series-parallel graphs of bounded depth. Both results extend the known class of graphs that have a single-logarithmic approximation ratio.