Node-Weighted Multicut in Planar Digraphs

TL;DR

This paper extends the KS22 result for Multicut in directed planar graphs from edge-weighted to node-weighted instances, achieving a deterministic $O(\log^2 n)$-approximation via the natural LP relaxation. The approach replaces KS22's randomized components with a deterministic region-growing technique and leverages layering and planar separators to decompose the problem into manageable δ-bounded minors, enabling a recursive cutting strategy around separator paths. A key contribution is showing that node-weights can be handled without a black-box reduction that preserves planarity, and the analysis yields a clean cost bound and feasibility. As a corollary, a standard reduction from Sparsest Cut gives an $O(\log^3 n)$-approximation for node-weighted Sparsest Cut in directed planar graphs, suggesting broader applicability to minor-free settings and potential extensions to polymatroid networks.

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 $\tildeΩ(n^{1/7})$ 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.

Figures (2)

• Figure 1: Example of consecutive layers constructed by \ref{['algo:layering']}, here $j$ is odd. Some boundary edges are shown with corresponding boundary nodes. The red boundary nodes are the ones included in $S$; thus the corresponding incident edges (shown with dashed red lines) are removed in $G \setminus S$.
• Figure 3: Here $Q$ is a $u$-$w$ path shown with a dashed blue line. All regions drawn are in-balls; we omit the out-balls for visual clarity as the analysis is similar. In this example, $b \leq_P a$ to illustrate that $a$ and $b$ can appear in any order, even though $v_{in}(a) \leq_P v_{in}(b)$ necessarily.

Theorems & Definitions (33)

• remark 1
• theorem 1: KS22
• corollary 2: KS22
• theorem 3
• corollary 4
• lemma 4
• definition 1
• lemma 5
• claim 6
• proof