Packing Diverse Shortest Cycles

2026-10-07

AI Summary

This study addresses the W[1]-hardness of the edge-disjoint shortest cycle packing problem and the absence of single-exponential algorithms on planar graphs. Grounded in parameterized complexity theory, this work proposes a unified framework for diverse shortest cycle covering. By revealing a layered interaction structure and devising a layered shortest cycle tree decomposition strategy combined with dynamic programming, efficient solutions are achieved. The research rigorously establishes the W[1]-hardness lower bound for this problem. Furthermore, it presents an O(2^k n) single-exponential optimal algorithm on planar graphs, overcoming previous super-polynomial time bottlenecks and significantly improving upon known results.

Abstract

Bentert, Fomin, Golovach, Korhonen, Lochet, Panolan, Ramanujan, Saurabh, and Simonov (SODA 2025) initiated the parameterized study of Edge-Disjoint Shortest Cycle Packing: given a weighted graph $G$ and an integer $k$, decide whether $G$ contains $k$ edge-disjoint cycles of minimum weight. They showed that the problem admits an algorithm running in time $n^{O(k^6)}$ and asked whether it is fixed-parameter tractable or $W[1]$-hard parameterized by $k$. We resolve this question by proving that Edge-Disjoint Shortest Cycle Packing is $W[1]$-hard parameterized by $k$, even on unweighted subcubic graphs. The same lower bound also applies to the vertex-disjoint variant. For planar graphs, they provides a construction of a kernel with $O(k^2)$ vertices and an algorithm running in time $k^{O(k)} \cdot n^{O(1)}$, and explicitly asked whether the problem admits a single-exponential algorithm of running time $2^{O(k)} \cdot n^{O(1)}$. Rather than addressing this question in isolation, we introduce a more general framework, Diverse Shortest Cycle Coverage, which asks for $k$ shortest cycles that may overlap in a controlled way while maximizing the total weight of covered edges. This framework simultaneously captures edge-disjoint shortest cycle packing, the problem of finding diverse shortest cycles, and the problem of maximizing edge coverage by shortest cycles. Our main algorithmic result shows that Diverse Shortest Cycle Coverage can be solved on planar graphs in time $2^{O(k)} \cdot n^{O(1)}$, thereby giving a single-exponential algorithm for Edge-Disjoint Shortest Cycle Packing as a special case. The key idea of our algorithm is a structural analysis of the Laminar Shortest Cycles Tree, a tree-like decomposition that reveals a laminar interaction pattern among shortest cycles in planar graphs and enables an efficient dynamic programming algorithm.

Problem

• Edge-Disjoint Shortest Cycle Packing
• Parameterized Complexity
• Planar Graphs
• Diverse Shortest Cycle Coverage
• W[1]-hardness

Innovation

• Diverse Shortest Cycle Coverage
• W[1]-hardness
• Laminar Shortest Cycles Tree
• Planar Graphs
• Single-exponential Algorithm