Erdös-Pósa property and its algorithmic applications: parity constraints, subset feedback set, and subset packing

The well-known Erdős-Pósa theorem says that for any integer k and any graph G, either G contains k vertexdisjoint cycles or a vertex setX of order at most c·k log k (for some constant c) such that G−X is a forest. Thomassen [39] extended this result to the even cycles, but on the other hand, it is well-known that this theorem is no longer true for the odd cycles. However, Reed [31] proved that this theorem still holds if we relax k vertex-disjoint odd cycles to k odd cycles with each vertex in at most two of them. These theorems initiate many researches in both graph theory and theoretical computer science. In the graph theory side, our problem setting is that we are given a graph and a vertex set S, and we want to extend all the above results to cycles that are required to go through a subset of S, i.e., each cycle contains at least one vertex in S (such a cycle is called an S-cycle). It was shown in [20] that the above Erdős-Pósa theorem still holds for this subset version. In this paper, we extend both Thomassen’s result and Reed’s result in this way. In the theoretical computer science side, we investigate generalizations of the following well-known problems in the framework of parameterized complexity: the feedback set problem and the cycle packing problem. Our purpose here is to consider the following problems: the feedback set problem with respect to the S-cycles, and the S-cycle packing problem. We give the first fixed parameter algorithms for the two problems. Namely; 1. For fixed k, we can either find a vertex set X of size k such that G − X has no S-cycle, or conclude that such a vertex set does not exist in O(nm) time (independently obtained in [7]). 2. For fixed k, we can either find k vertex-disjoint S-cycles, or conclude that such k disjoint cycles do not exist in O(nm) time. We also extend the above results to those with the parity constraints as follows; 1. For a parameter k, there exists a fixed parameter algorithm that either finds a vertex set X of size k such that G−X has no even S-cycle, or concludes that such a vertex set does not exist. 2. For a parameter k, there exists a fixed parameter algorithm that either finds a vertex set X of size k such ∗University of Tokyo, Tokyo 113-8656, Japan. Partly supported by Grant-in-Aid for Scientific Research and by Global COE Program “The research and training center for new development in mathematics,” MEXT, Japan. E-mail: {kakimura, kobayashi}@mist.i.u-tokyo.ac.jp †National Institute of Informatics, 101-8430, Japan. Partly supported by JSPS, Grant-in-Aid for Scientific Research, by C & C Foundation, and by Inoue Research Award for Young Scientists. Email: k [email protected] that G−X has no odd S-cycle, or concludes that such a vertex set does not exist. 3. For a parameter k, there exists a fixed parameter algorithm that either finds k vertex-disjoint even Scycles, or concludes that such k disjoint cycles do not exist. 4. For a parameter k, there exists a fixed parameter algorithm that either finds k odd S-cycles with each vertex in at most two of them, or concludes that such k cycles do not exist.