TY - JOUR
TI - An O(log OPT)-Approximation for Covering and Packing Minor Models of θr
AU - Chatzidimitriou, D.
AU - Raymond, J.-F.
AU - Sau, I.
AU - Thilikos, D.M.
JO - Algorithmica (New York)
PY - 2018
VL - 80
TODO - 4
SP - 1330-1356
PB - Springer New York LLC
SN - null
TODO - 10.1007/s00453-017-0313-5
TODO - Approximation algorithms;  Graphic methods;  Machine design;  Machinery, Covering and packing;  Disjoint subgraphs;  NP-hard;  Positive integers;  Two-graphs;  Unified approach, Graph theory
TODO - Given two graphs G and H, we define v- coverH(G) (resp. e- coverH(G)) as the minimum number of vertices (resp. edges) whose removal from G produces a graph without any minor isomorphic to H. Also v- packH(G) (resp. e- packH(G)) is the maximum number of vertex- (resp. edge-) disjoint subgraphs of G that contain a minor isomorphic to H. We denote by θr the graph with two vertices and r parallel edges between them. When H= θr, the parameters v- coverH, e- coverH, v- packH, and e- packH are NP-hard to compute (for sufficiently big values of r). Drawing upon combinatorial results in Chatzidimitriou et al. (Minors in graphs of large θr-girth, 2015, arXiv:1510.03041), we give an algorithmic proof that if v-packθr(G)≤k, then v-coverθr(G)=O(klogk), and similarly for e-packθr and e-coverθr. In other words, the class of graphs containing θr as a minor has the vertex/edge Erdős–Pósa property, for every positive integer r. Using the algorithmic machinery of our proofs we introduce a unified approach for the design of an O(log OPT) -approximation algorithm for v-packθr, v-coverθr, e-packθr, and e-coverθr that runs in O(n· log (n) · m) steps. Also, we derive several new Erdős–Pósa-type results from the techniques that we introduce. © 2017, The Author(s).
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