A matching in a graph is induced if no two of its edges are joined by an edge, and finding a large induced matching is a very hard problem. Lin et al. (2018) provide an approximation algorithm with ratio $\Delta$ for the weighted version of the induced matching problem on graphs of maximum degree $\Delta$. Their approach is based on an integer linear programming formulation whose integrality gap is at least $\Delta -1$, that is, their approach offers only little room for improvement in the weighted case. For the unweighted case though, we conjecture that the integrality gap is at most $\frac{5}{8}\Delta +O\left(1\right)$, and that also the approximation ratio can be improved at least to this value. We provide primal–dual approximation algorithms with ratios $(1-ϵ)\Delta +\frac{1}{2}$ for general $\Delta$ with $ϵ\approx 0.02005$, and $\frac{7}{3}$ for $\Delta =3$. Furthermore, we prove a best-possible bound on the fractional induced matching number in terms of the order and the maximum degree.

如果图中没有两个边被一条边连接，就会引起图中的匹配，而找到一个大的诱导匹配是一个非常困难的问题。Lin等。（2018）提供具有比率的近似算法$\Delta$ 最大度图上的诱导匹配问题的加权形式 $\Delta$。他们的方法基于整数线性规划公式，其积分间隙至少为$\Delta -\mathrm{1个}$，也就是说，他们的方法在加权情况下几乎没有改善的余地。但是对于未加权的情况，我们推测完整性差距最大为$\frac{5}{8}\Delta +Ø（\mathrm{1个}）$，并且近似率也可以至少提高到该值。我们提供具有比率的原始对偶逼近算法$（\mathrm{1个}-ϵ）\Delta +\frac{\mathrm{1个}}{2}$ 对于一般 $\Delta$ 与 $ϵ\approx 0。02005$和 $\frac{7}{3}$ 对于 $\Delta =3$。此外，我们证明了在分数诱导匹配数上的最大可能阶的顺序和最大程度。