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 round-and-bipartize 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/ρ, where 2ρ − 1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph G̃:= G/S and satisfies ρ ∈ 2, ∞, with ρ = ∞ corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1 + 1/ρ) (1 − α) + 2α, where α ∈ 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 G̃, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also 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, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ρ and α, which are ρ = 2 and α = 1 − 4/n, recover the integrality gap of 2 − 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph.
Kashaev et al. (Mon,) studied this question.