Abstract
A key result in the field of kernelization, a subfield of parameterized complexity, states that the classic Disjoint Cycle Packing problem, i.e., finding $k$ vertex disjoint cycles in a given graph...
Highlights
Polynomial-time preprocessing is one of the widely used methods to tackle NP-hard problems in practice, as it plays well with exact algorithms, heuristics, and approximation algorithms
Each problem instance is coupled with a parameter k and the parameterized problem is said to admit a kernel if there is a polynomial-time algorithm, called a kernelization algorithm, that reduces the input instance down to an instance whose size is bounded by a function f (k) in k, while preserving the answer
For several important problems, there are settings in which we need not be very strict about constraints. This is interesting for “strict” problems where, e.g., (a) it is known that no polynomial kernels are possible unless NP ⊆ coNP/poly, or where (b) the algorithm with the best running time matches 26:4 Kernelization of Cycle Packing with Relaxed Disjointness Constraints the known lower bound, or where (c) no considerable improvements have been made either algorithmically or in terms of kernel upper/lower bounds
Summary
Polynomial-time preprocessing is one of the widely used methods to tackle NP-hard problems in practice, as it plays well with exact algorithms, heuristics, and approximation algorithms. There exists a constant c such that every (multi) graph either contains k vertex disjoint cycles or it has a feedback vertex set of size at most ck log k.
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