  We study the Maximum Budgeted Allocation problem, i.e., the problem of
selling a set of $m$ indivisible goods to $n$ players, each with a separate
budget, such that we maximize the collected revenue. Since the natural
assignment LP is known to have an integrality gap of $\frac{3}{4}$, which
matches the best known approximation algorithms, our main focus is to improve
our understanding of the stronger configuration LP relaxation. In this
direction, we prove that the integrality gap of the configuration LP is
strictly better than $\frac{3}{4}$, and provide corresponding polynomial time
roundings, in the following restrictions of the problem: (i) the Restricted
Budgeted Allocation problem, in which all the players have the same budget and
every item has the same value for any player it can be sold to, and (ii) the
graph MBA problem, in which an item can be assigned to at most 2 players.
Finally, we improve the best known upper bound on the integrality gap for the
general case from $\frac{5}{6}$ to $2\sqrt{2}-2\approx 0.828$ and also prove
hardness of approximation results for both cases.
