The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a $2$-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a simple rounding algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair $(G, S)$, consisting of a graph with an odd cycle transversal. If $S$ is a stable set, we prove a tight approximation ratio of $1 + 1/\rho$, where $2\rho -1$ denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph $\tilde{G} := G /S$ and satisfies $\rho \in [2,\infty]$. If $S$ is an arbitrary set, we prove a tight approximation ratio of $\left(1+1/\rho \right) (1 - \alpha) + 2 \alpha$, where $\alpha \in [0,1]$ is a natural parameter measuring the quality of the set $S$. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph $\tilde{G}$. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals and show optimality of the analysis.

翻译：顶点覆盖问题是一个基础且被广泛研究的组合优化问题。已知其标准线性规划松弛在二分图上具有整数性，在一般图上具有半整数性。因此，基于该松弛的自然舍入算法可在二分图上计算最优解，并在一般图上得到2-近似解。这引发了一个问题：能否根据图与二分图的接近程度，以超越最坏情况的方式插值标准线性规划松弛的舍入曲线？本文考虑一种简单的舍入算法，该算法利用诱导二分子图的知识来获得改进的近似比。等价地，我们假设处理一个由图和奇环横贯组成的对$(G, S)$。若$S$为稳定集，我们证明紧近似比为$1 + 1/\rho$，其中$2\rho -1$表示收缩图$\tilde{G} := G /S$的奇围长（即最短奇环长度），且满足$\rho \in [2,\infty]$。若$S$为任意集合，我们证明紧近似比为$\left(1+1/\rho \right) (1 - \alpha) + 2 \alpha$，其中$\alpha \in [0,1]$是衡量集合$S$质量的自然参数。证明紧改进近似比的关键技术依赖于对收缩图$\tilde{G}$的结构分析。通过构造匹配所得上界的权函数类来证明紧性。作为结构分析的副产品，我们得到了3-可着色图的积分间隙和分数色数的改进紧界。我们还讨论了寻找优质奇环横贯的算法应用，并证明了分析的最优性。