Packing Short Cycles
Matthias Bentert, Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen,, William Lochet, Fahad Panolan, M. S. Ramanujan, Saket Saurabh, and Kirill, Simonov

TL;DR
This paper studies cycle packing problems in graphs, providing polynomial algorithms for fixed k, establishing W[1]-hardness, and demonstrating fixed-parameter tractability and kernelization results for special graph classes.
Contribution
It introduces polynomial algorithms for Min-Sum Cycle Packing with fixed k, proves W[1]-hardness, and shows FPT and kernelization results for Shortest Cycle Packing on planar graphs.
Findings
Polynomial-time algorithm for fixed k Min-Sum Cycle Packing
W[1]-hardness of Min-Sum Cycle Packing parameterized by k
FPT algorithm and polynomial kernel for Shortest Cycle Packing on planar graphs
Abstract
Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Bj\"orklund, Husfeldt, ICALP 2014; Mari, Mukherjee, Pilipczuk, and Sankowski, SODA 2024] or request the paths to be shortest [Lochet, SODA 2021], we consider the following cycle packing problems: Min-Sum Cycle Packing and Shortest Cycle Packing. In Min-Sum Cycle Packing, we try to find, in a weighted undirected graph, vertex-disjoint cycles of minimum total weight. Our first main result is an algorithm that, for any fixed , solves the problem in polynomial time. We complement this result by establishing the W[1]-hardness of Min-Sum Cycle Packing parameterized by . The same results hold for the version of the problem where the task is to…
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Taxonomy
TopicsOptimization and Packing Problems
