  We study the basic allocation problem of assigning resources to players so as
to maximize fairness. This is one of the few natural problems that enjoys the
intriguing status of having a better estimation algorithm than approximation
algorithm. Indeed, a certain Configuration-LP can be used to estimate the value
of the optimal allocation to within a factor of $4 + {\epsilon}$. In contrast,
however, the best known approximation algorithm for the problem has an
unspecified large constant guarantee.
  In this paper we significantly narrow this gap by giving a $13$-approximation
algorithm for the problem. Our approach develops a local search technique
introduced by Haxell [Hax95] for hypergraph matchings, and later used in this
context by Asadpour, Feige, and Saberi [AFS12]. For our local search procedure
to terminate in polynomial time, we introduce several new ideas such as lazy
updates and greedy players. Besides the improved approximation guarantee, the
highlight of our approach is that it is purely combinatorial and uses the
Configuration-LP only in the analysis.
