In the Graph bipartization (or Odd Cycle Transversal) problem, the objective is to decide whether a given graph G can be made bipartite by the deletion of k vertices for some given k. The parameterized complexity of Odd Cycle Transversal was resolved in the breakthrough paper of Reed, Smith and Vetta [Operations Research Letters, 2004], who developed an algorithm running in time $\mathcal{O}(3^{k}kmn)$. The question of improving the dependence on the input size to linear, which was another long standing open problem in the area, was resolved by Iwata et al. [SICOMP 2016] and Ramanujan and Saurabh [TALG 2017], who presented $\mathcal{O}(4^k(m+n))$ and $4^k{k}^{\mathcal{O}(1)}(m+n)$ algorithms respectively. In this paper, we obtain a faster algorithm that runs in time $3^k{k}^{\mathcal{O}(1)}(m+n)$ and hence preserves the linear dependence on the input size while nearly matching the dependence on k incurred by the algorithm of Reed, Smith and Vetta.

在图二分图（或奇数周期遍历）问题中，目标是确定是否可以通过删除某些给定k的k个顶点来使给定图G成为二分图。Reed，Smith和Vetta的突破性论文[Operations Research Letters，2004]解决了奇数循环遍历的参数化复杂性问题，该论文开发了及时运行的算法$\mathcal{Ø}（3^{ķ}ķ米ñ）$。Iwata等人解决了改善对线性输入大小的依赖性的问题，这是该地区另一个长期存在的开放性问题。[SICOMP 2016]和Ramanujan和Saurabh [TALG 2017]$\mathcal{Ø}（4^{ķ}（米+ñ））$ 和 $4^{ķ}{ķ}^{\mathcal{Ø}（\mathrm{1个}）}（米+ñ）$算法分别。在本文中，我们获得了一种可以及时运行的更快算法$3^{ķ}{ķ}^{\mathcal{Ø}（\mathrm{1个}）}（米+ñ）$因此，保留了对输入大小的线性依赖关系，同时几乎与Reed，Smith和Vetta算法对k的依赖关系相匹配。