(Almost-)Optimal FPT Algorithm and Kernel for $T$-Cycle on Planar Graphs

TL;DR

This work studies the T-Cycle problem on planar graphs, where one seeks a simple cycle through a given terminal set T. It delivers a subexponential fixed-parameter tractable algorithm with running time $2^{O(\sqrt{k}\log k)}\cdot n$ and a linear-time kernel of size $k\log^{O(1)}k$, both optimal up to polylogarithmic factors under ETH. The algorithmic core combines a treewidth reduction via irrelevant-vertex removal (leading to $\mathrm{tw}=O(\sqrt{k}\log k)$), CL-configurations with segment-type analysis, Reed’s plane-cutting technique for linear-time preprocessing, and protrusion-based kernelization augmented by B-linkage equivalence for Disjoint Paths. The results establish the first subexponential FPT algorithm and the first polynomial kernel for T-Cycle on planar graphs, and introduce novel techniques (nested protrusions, segment forests) that may extend to broader problems. Together, they advance the understanding of terminal-based routing in planar graphs and provide robust preprocessing and DP-tools that could inform similar parameterized problems.

Abstract

Research of cycles through specific vertices is a central topic in graph theory. In this context, we focus on a well-studied computational problem, \textsc{$T$-Cycle}: given an undirected $n$-vertex graph $G$ and a set of $k$ vertices $T\subseteq V(G)$ termed \textit{terminals}, the objective is to determine whether $G$ contains a simple cycle $C$ through all the terminals. Our contribution is twofold: (i) We provide a $2^{O(\sqrt{k}\log k)}\cdot n$-time fixed-parameter deterministic algorithm for \textsc{$T$-Cycle} on planar graphs; (ii) We provide a $k^{O(1)}\cdot n$-time deterministic kernelization algorithm for \textsc{$T$-Cycle} on planar graphs where the produced instance is of size $k\log^{O(1)}k$. Both of our algorithms are optimal in terms of both $k$ and $n$ up to (poly)logarithmic factors in $k$ under the ETH. In fact, our algorithms are the first subexponential-time fixed-parameter algorithm for \textsc{$T$-Cycle} on planar graphs, as well as the first polynomial kernel for \textsc{$T$-Cycle} on planar graphs. This substantially improves upon/expands the known literature on the parameterized complexity of the problem.

Figures (20)

• Figure 1: (a) A set $\mathcal{C}$ of concentric cycles and $v$ is $r$-isolated. (b) A CL-configuration $(\mathcal{C},L)$. The segments of $L$ are represented in green and the edges of $L$ outside of $D_r$ are shown in blue. Hence, the green edges along with the blue edges combine to give the loop $L$. In both subfigures, the terminal vertices are highlighted in red.
• Figure 2: (a) $S_1$ and $S_2$ are $C_j$-segments with endpoints $u_1,v_1$ and $u_2,v_2$, respectively. Moreover, $P$ and $P'$ are the $(u_1,u_2)$-path and $(v_1,v_2)$-path along $C_j$, respectively. Here, $S_1$ and $S_2$ have the same $C_j$-type. (b) Here, segments $S_1,\ldots,S_{10}$ are $C_j$-segments such that segment $S_i$ has endpoints $u_i$ and $v_i$. Segments $S_4$ and $S_7$ are not of the same $C_j$-type because of condition (3) in Definition \ref{['D:ST']}, and $S_6$ and $S_2$ are not of the same $C_j$-type because of condition (2) in Definition \ref{['D:ST']}. Further, when $S_6$ and $S_2$ are restricted to $D_{j-2}$, they have the same $C_{j-2}$-type. Finally, $S_{10}\prec S_9 \prec S_8 \prec S_7$, and similarly, $S_6\prec S_5\prec S_4$.
• Figure 3: (a) Here, $S_i$ is the $C_j$ segment with endpoints $u_i$ and $v_i$, segments $S_1,S_2,S_3$ are illustrated in red, and segments $S_1,\ldots,S_8$ have the same $C_j$-type. (b) In the rerouting, we remove the segment $S_2$ completely (reducing the cost) and use a part of the segment $S_1$ (not increasing the cost), and additionally use some paths along concentric cycles $C_j$ and $C_{j-1}$ that have cost 0.
• Figure 4: Subfigure (a) depicts $C_j$ segments of a CL configuration $\mathcal{Q}$ of depth $r$. Here, segments $S_1,\ldots,S_{10}$ are $C_j$-segments such that segment $S_i$ has endpoints $u_i$ and $v_i$. All $C_j$-segments of the same type are depicted in a common color. The colors red, blue green and yellow are used to distinguish segment types. Subfigure (b) shows the $j$-th segment forest of $\mathcal{Q}$.
• Figure 5: A $2$-punctured plane $\boxdot$ is depicted (in blue) by its two holes highlighted in bold. $v$ is a vertex in $\boxdot$ which is $5g(k)+1$-isolated in $G$, and hence there is a sequence of $C_0,\ldots,C_{5g(k)+1}$ concentric cycles separating $v$ and $T$. Since $v$ is not $4g(k)$-isolated from boundary of $\boxdot$, there is some boundary vertex $w$ of $\boxdot$ such that $w\in D_{4g(k)}$. In this case, a sequence of concentric cycles $C_{4g(k)+1},\ldots,C_{5g(k)+1}$ separate $w$ from $T$, implying that $w$ is $g(k)$-isolated.

Theorems & Definitions (79)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Lemma 5: Informal Statement of \ref{['L:main']}
• Lemma 6: Informal Statement of \ref{['thm:logarithmic']}
• Definition 7: Segment types
• Proposition 8: Lemma 2 in reedLinear
• Definition 9: Treewidth
• Proposition 10: JCTB