A Round and Bipartize Approximation Algorithm for Vertex Cover

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 obtain improved bounds on the approximation ratio, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we have access to a subset of vertices $S$ whose removal bipartizes the graph. If $S$ is an independent set, we prove an approximation ratio of $1 + 1/\rho$, where $2\rho -1$ denotes the odd girth of the contracted graph $\tilde{\mathcal{G}} := \mathcal{G} /S$ and thus satisfies $\rho \geq 2$. We show that this is tight for any graph and independent set by providing a family of weight functions for which this bound is attained. In addition, we give tight upper bounds for the fractional chromatic number and the integrality gap of such graphs, both of which also depend on the odd girth. If $S$ is an arbitrary set, we prove a tight approximation ratio of $\left(1+1/\rho \right) (1 - \alpha) + 2 \alpha$, where $\alpha \in [0,1]$ denotes the total normalized dual sum of the edges lying inside of the set $S$. As an algorithmic application, we show that for any efficiently $k$-colorable graph with $k \geq 4$ we can find a bipartizing set satisfying $\alpha \leq 1 - 4/k$. This provides an approximation algorithm recovering the bound of $2 - 2/k$ in the worst case (i.e., when $\rho = 2$), which is best possible for this setting when using the standard relaxation.