  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.
(Approximating weighted induced matchings, Discrete Applied Mathematics 243
(2018) 304-310) 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
$\frac{5}{8}\Delta+O(1)$, and that also the approximation ratio can be improved
at least to this value. We provide primal-dual approximation algorithms with
ratios $(1-\epsilon) \Delta + \frac{1}{2}$ for general $\Delta$ with $\epsilon
\approx 0.02005$, 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.
