Packing Short Cycles
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 find edge-disjoint cycles. Our second main result concerns Shortest Cycle Packing, which is a special case of Min-Sum Cycle Packing that asks to find a packing of shortest cycles in a graph. We prove this problem to be fixed-parameter tractable (FPT) when parameterized by on weighted planar graphs. We also obtain a polynomial kernel for the edge-disjoint variant of the problem on planar graphs. Deciding whether Min-Sum Cycle Packing is FPT on planar graphs and whether Shortest Cycle Packing is FPT on general graphs remain challenging open questions.
Cite
@article{arxiv.2410.18878,
title = {Packing Short Cycles},
author = {Matthias Bentert and Fedor V. Fomin and Petr A. Golovach and Tuukka Korhonen and William Lochet and Fahad Panolan and M. S. Ramanujan and Saket Saurabh and Kirill Simonov},
journal= {arXiv preprint arXiv:2410.18878},
year = {2024}
}