Node-Weighted Multicut in Planar Digraphs

TL;DR

This paper extends an $O( ext{log}^2 n)$-approximation algorithm for Multicut in planar digraphs to the node-weighted case, providing a deterministic solution and simplifying the original analysis.

Contribution

It introduces a deterministic algorithm for node-weighted Multicut in planar digraphs, extending previous work and clarifying the analysis.

Findings

Achieved a deterministic $O( ext{log}^2 n)$-approximation for node-weighted Multicut in planar digraphs.

Simplified and clarified the algorithm and analysis from previous randomized methods.

Implication for approximating Nonuniform Sparsest Cut with an additional logarithmic factor.

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.