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 and Husfeldt, ICALP 2014; Mari et al., 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. Whether Min-Sum Cycle Packing is FPT on planar graphs, or Shortest Cycle Packing on general graphs, remains open.