  Asadpour, Feige, and Saberi proved that the integrality gap of the
configuration LP for the restricted max-min allocation problem is at most $4$.
However, their proof does not give a polynomial-time approximation algorithm. A
lot of efforts have been devoted to designing an efficient algorithm whose
approximation ratio can match this upper bound for the integrality gap. In
ICALP 2018, we present a $(6 + \delta)$-approximation algorithm where $\delta$
can be any positive constant, and there is still a gap of roughly $2$. In this
paper, we narrow the gap significantly by proposing a
$(4+\delta)$-approximation algorithm where $\delta$ can be any positive
constant. The approximation ratio is with respect to the optimal value of the
configuration LP, and the running time is $\mathit{poly}(m,n)\cdot
n^{\mathit{poly}(\frac{1}{\delta})}$ where $n$ is the number of players and $m$
is the number of resources. We also improve the upper bound for the integrality
gap of the configuration LP to $3 + \frac{21}{26} \approx 3.808$.
