Packing Short Cycles

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 [Björklund, 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, k vertex-disjoint cycles of minimum total weight. Our first main result is an algorithm that, for any fixed k, solves the problem in polynomial time. We complement this result by establishing the W[1]-hardness of Min-Sum Cycle Packing parameterized by k. The same results hold for the version of the problem where the task is to find k 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 k shortest cycles in a graph. We prove this problem to be fixed-parameter tractable (FPT) when parameterized by k 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.