Abstract
The induced odd cycle packing number iocp(G) of a graph G is the maximum integer k such that G contains an induced subgraph consisting of k pairwise vertex-disjoint odd cycles. Motivated by applications to geometric graphs, Bonamy et al. proved that graphs of bounded induced odd cycle packing number, bounded Vapnik-Chervonenkis (VC) dimension, and linear independence number admit a randomized efficient polynomial-time approximation scheme for the independence number.
We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm that for any integers k >= 0 and t >= 1 and any n-vertex graph G of induced odd cycle packing number at most k returns in time O-k,O-t(n(k+4)) an independent set of G whose size is at least alpha(G)-n/t (G)-n with high probability. In addition, we present chi-boundedness results for graphs with bounded odd cycle packing number, and use them to design a quasipolynomial-time approximation scheme for the independence number only assuming bounded induced odd cycle packing number.