Περίληψη:
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).
Συγγραφείς:
Chatzidimitriou, D.
Raymond, J.-F.
Sau, I.
Thilikos, D.M.
Λέξεις-κλειδιά:
Approximation algorithms; Graphic methods; Machine design; Machinery, Covering and packing; Disjoint subgraphs; NP-hard; Positive integers; Two-graphs; Unified approach, Graph theory
