ETH-Tight Algorithm for Cycle Packing on Unit Disk Graphs

TL;DR

This work tackles Planar Disjoint Paths by combining treewidth reduction, a 1-punctured-plane decomposition, and a ring/frame framework to constrain how multiple $s_i$–$t_i$ paths interact. It builds a canonical weak linkage via discrete homotopy and a crossing-pattern encoding against frames, enabling a structured recovery of a real linkage through a final application of Schrijver’s planar-disjoint-paths machinery. The main result is an algorithm running in $2^{O(k^2)}n$ time, with an ETH-aligned lower bound supporting the claimed optimality; the approach advances previous bounds and introduces a modular, geometrically grounded toolkit (rings, frames, abstract polygonal schemas) that may be of independent use for planar routing problems. The combination of treewidth reduction, ring-based decomposition, and homology-informed recovery offers a robust framework for single-exponential-in-$k$ algorithms in planar graph routing tasks. Overall, the paper delivers a near-optimal parameterized algorithm for Planar Disjoint Paths and contributes a transferable set of techniques for handling complex path systems in planar geometries.

Abstract

In this paper, we consider the Cycle Packing problem on unit disk graphs defined as follows. Given a unit disk graph G with n vertices and an integer k, the goal is to find a set of $k$ vertex-disjoint cycles of G if it exists. Our algorithm runs in time $2^{O(\sqrt k)}n^{O(1)}$. This improves the $2^{O(\sqrt k\log k)}n^{O(1)}$-time algorithm by Fomin et al. [SODA 2012, ICALP 2017]. Moreover, our algorithm is optimal assuming the exponential-time hypothesis.

Figures (8)

• Figure 1: (a)
• Figure 2: (a)
• Figure 3: (a) The vertices of $I_0, I_1$ and $I_2$ are marked with disks. (b) This sequence of concentric cycles is not tight. The witness of non-tightness is the path colored gray. The path violates the third condition of the tightness. (c) A path in $\mathcal{P}$ connecting $s$ and $t$ has four segments in $\textsf{Ring}(i,j)$: the outer visitor connects $a_1$ and $a_2$, the inner visitor connects $b_2$ and $b_3$, and the two traversing segments connecting $b_1$ and $a_3$, and $a_4$ and $b_4$, respectively.
• Figure 4: (a) $B_2$ is the union of $Q_2$ and $C_{\alpha+2}$, which is the subgraph of $G$ colored gray. (b) $S^o$ is the noose colored gray. It passes through at most $2^{2k}$ vertices of $G$. An inner visitor $\gamma$ (colored red) has two endpoints $a$ and $b$, and its base arc is colored blue.
• Figure 5: (a) The red path $\gamma$ is an inner visitor. If its order is zero, no segment of $\mathcal{P}$ intersecting $S^o$ has endpoints on the base of $\gamma$. Therefore, the blue path does not intersect any other segments of $\mathcal{P}$, and thus we can replace the red path with the blue path. (b) $S$ is a traversing segment. The subpath of $S$ lying between $a$ and $b$ is an $r$-horn. The base of the $r$-horn is colored blue.

Theorems & Definitions (35)

• Theorem 1
• Theorem 2
• Lemma 3: Lemmas 1 and 10 in adler2017irrelevant
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• Lemma 7: Lemma 2 in reed1995rooted
• Lemma 8