Abstract
The Cycle Packing problem asks whether a given undirected graph $G=(V,E)$ contains $k$ vertex-disjoint cycles. Since the publication of the classic Erdos--Posa theorem in 1965, this problem receive...
Highlights
The Cycle Packing problem asks whether a given undirected graph G = (V, E) contains k vertex-disjoint cycles
Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 71:2 Packing Cycles Faster Than Erdős-Pósa associated with a parameter k that is a non-negative integer, and a problem is said to be fixed-parameter tractable (FPT) if the combinatorial explosion in the time complexity can be confined to the parameter k
The Erdős-Pósa theorem states that there exists a function f (k) = O(k log k) such that for each non-negative integer k, every undirected graph either contains k vertex-disjoint cycles or it has a feedback vertex set consisting of f (k) vertices [15]
Summary
The Erdős-Pósa theorem states that there exists a function f (k) = O(k log k) such that for each non-negative integer k, every undirected graph either contains k vertex-disjoint cycles or it has a feedback vertex set consisting of f (k) vertices [15]. It is well known that the treewidth (tw) of a graph is not larger than its feedback vertex set number (fvs), and that a naive dynamic programming (DP) scheme solves Cycle Packing in time 2O(tw log tw) · |V | and exponential space (see, e.g., [10]). Pósa [15] proved that the bound f (k) = O(k log k) in their theorem is essentially tight as there exist infinitely many graphs and a constant c such that these graphs do not contain k vertex-disjoint cycles and yet their feedback vertex set number is larger than ck log k. Cygan et al [11] proved that the bound 2O(tw log tw) · |V |O(1) is likely to be essentially tight in the sense that unless the Exponential Time Hypothesis (ETH) [20] is false, Cycle Packing cannot be solved in time 2o(tw log tw) · |V |O(1) (it might still be true that Cycle Packing is solvable in time 2o(fvs log fvs) · |V |O(1))
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