  The discrete $\alpha$-neighbor $p$-center problem (d-$\alpha$-$p$CP) is an
emerging variant of the classical $p$-center problem which recently got
attention in literature. In this problem, we are given a discrete set of points
and we need to locate $p$ facilities on these points in such a way that the
maximum distance between each point where no facility is located and its
$\alpha$-closest facility is minimized. The only existing algorithms in
literature for solving the d-$\alpha$-$p$CP are approximation algorithms and
two recently proposed heuristics.
  In this work, we present two integer programming formulations for the
d-$\alpha$-$p$CP, together with lifting of inequalities, valid inequalities,
inequalities that do not change the optimal objective function value and
variable fixing procedures. We provide theoretical results on the strength of
the formulations and convergence results for the lower bounds obtained after
applying the lifting procedures or the variable fixing procedures in an
iterative fashion. Based on our formulations and theoretical results, we
develop branch-and-cut (B&C) algorithms, which are further enhanced with a
starting heuristic and a primal heuristic.
  We evaluate the effectiveness of our B&C algorithms using instances from
literature. Our algorithms are able to solve 116 out of 194 instances from
literature to proven optimality, with a runtime of under a minute for most of
them. By doing so, we also provide improved solution values for 116 instances.
