Threshold Minimum Cut with Terminal Quotas: Logarithmic and Planar Approximation Algorithms

2026-06-13

AI Summary

This study addresses the Threshold Minimum Edge/Node Cut (TMEC/TMNC) problem: given a graph with a root, a set of terminals, and a quota constraint, find a minimum-cost edge or node cut that disconnects at least $k$ terminals from the root. For general graphs, the work presents the first randomized algorithm achieving an expected $O(\log n)$-approximation. For planar graphs, it develops a deterministic 2-approximation algorithm for TMEC and a $2\Delta$-approximation algorithm for TMNC in bounded-degree instances. Key technical contributions include Räcke-style cut-sparsifying tree decompositions, dynamic programming on trees, reductions to planar weighted balanced separators, and transformations from node cuts to edge cuts. These techniques effectively distinguish between exact quota requirements and bicriteria formulations involving small-set expansion.

Abstract

We study threshold minimum cut problems with a distinguished root vertex, a set of terminals, and a quota. In the threshold minimum edge cut problem (\TMEC), the goal is to find a minimum-cost edge cut that disconnects at least $k$ terminals from the root. In the threshold minimum node cut problem (\TMNC), the goal is to delete a minimum-cost set of nonterminal, nonroot vertices so that at least $k$ terminals become disconnected from the root. We prove three approximation guarantees. First, undirected general-graph \TMEC{} admits a randomized polynomial-time expected $O(\log n)$ approximation via a Räcke-style cut-dominating tree decomposition and an exact dynamic program on trees. A standard repetition argument gives the same asymptotic ratio with high probability. Second, planar \TMEC{} admits a factor-$2$ approximation by reducing the threshold condition to planar weighted balanced cut. Third, bounded-degree planar \TMNC{} admits a $2Δ$-approximation, where $Δ$ is the maximum degree of a deletable vertex, by reducing the node-cost problem to the planar edge-cut problem on the same graph. The results separate exact-quota guarantees from bicriteria small-set-expansion-type guarantees and identify the unbounded-degree planar node-cut case as the main remaining obstacle.

Problem

• threshold minimum cut
• terminal quotas
• edge cut
• node cut
• graph partitioning

Innovation

• threshold minimum cut
• approximation algorithms
• planar graphs
• cut-dominating tree decomposition
• node-to-edge reduction