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

2025-04-27

AI Summary

This paper studies the T-Cycle problem on planar graphs: given a planar graph and a terminal set (T) of size (k), does there exist a simple cycle visiting all terminals? Addressing this long-standing open problem, we present the first subexponential-time FPT algorithm and the first polynomial kernel for planar graphs. Our approach deeply exploits structural properties of planar graphs, integrating divide-and-conquer, dynamic programming, grid-minor theory, and kernelization techniques to design deterministic algorithms. Our main contributions are: (1) an FPT algorithm running in time (2^{O(sqrt{k} log k)} cdot n), which is asymptotically optimal under the Exponential Time Hypothesis (ETH); and (2) the first deterministic polynomial kernel of size (k cdot log^{O(1)} k). Together, these results break the longstanding complexity barrier for the T-Cycle problem on planar graphs.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Kernel Methods

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
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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, extsc{$T$-Cycle}: given an undirected $n$-vertex graph $G$ and a set of $k$ vertices $Tsubseteq V(G)$ termed extit{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 extsc{$T$-Cycle} on planar graphs; (ii) We provide a $k^{O(1)}cdot n$-time deterministic kernelization algorithm for extsc{$T$-Cycle} on planar graphs where the produced instance is of size $klog^{O(1)}k$. 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 extsc{$T$-Cycle} on planar graphs, as well as the first polynomial kernel for extsc{$T$-Cycle} on planar graphs. This substantially improves upon/expands the known literature on the parameterized complexity of the problem.

Problem

• Develops optimal FPT algorithm for T-Cycle in planar graphs
• Provides polynomial kernel for T-Cycle on planar graphs
• Improves parameterized complexity results for T-Cycle problem

Innovation

• Optimal FPT algorithm for T-Cycle on planar graphs
• Polynomial kernelization for T-Cycle on planar graphs
• Subexponential-time deterministic algorithm for T-Cycle