Node-Weighted Multicut in Planar Digraphs

2026-01-27

AI Summary

This work proposes the first deterministic approximation algorithm for the node-weighted multiway cut problem in planar directed graphs, overcoming the technical barrier that planarity constraints impede the standard reduction of node weights to edge weights. Building upon a natural linear programming relaxation and leveraging structural properties of planar graphs together with a deterministic rounding scheme, the algorithm achieves an $O(\log^2 n)$ approximation ratio. The approach not only simplifies the intricate steps of prior randomized algorithms but also, via a standard reduction, yields an approximation guarantee for the non-uniform sparsest cut problem, albeit with an additional logarithmic factor loss in the approximation bound.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Kawarabayashi and Sidiropoulos [KS22] obtained an $O(\log^2 n)$-approximation algorithm for Multicut in planar digraphs via a natural LP relaxation, which also establishes a corresponding upper bound on the multicommodity flow-cut gap. Their result is in contrast to a lower bound of $\tilde{\Omega}(n^{1/7})$ on the flow-cut gap for general digraphs due to Chuzhoy and Khanna [CK09]. We extend the algorithm and analysis in [KS22] to the node-weighted Multicut problem. Unlike in general digraphs, node-weighted problems cannot be reduced to edge-weighted problems in a black box fashion due to the planarity restriction. We use the node-weighted problem as a vehicle to accomplish two additional goals: (i) to obtain a deterministic algorithm (the algorithm in [KS22] is randomized), and (ii) to simplify and clarify some aspects of the algorithm and analysis from [KS22]. The Multicut result, via a standard technique, implies an approximation for the Nonuniform Sparsest Cut problem with an additional logarithmic factor loss.

Problem

• Node-Weighted Multicut
• Planar Digraphs
• Multicommodity Flow-Cut Gap
• Nonuniform Sparsest Cut

Innovation

• node-weighted Multicut
• planar digraphs
• deterministic approximation algorithm
• flow-cut gap
• Nonuniform Sparsest Cut