  We study local search algorithms for metric instances of facility location
problems: the uncapacitated facility location problem (UFL), as well as
uncapacitated versions of the $k$-median, $k$-center and $k$-means problems.
All these problems admit natural local search heuristics: for example, in the
UFL problem the natural moves are to open a new facility, close an existing
facility, and to swap a closed facility for an open one; in $k$-medians, we are
allowed only swap moves. The local-search algorithm for $k$-median was analyzed
by Arya et al. (SIAM J. Comput. 33(3):544-562, 2004), who used a clever
``coupling'' argument to show that local optima had cost at most constant times
the global optimum. They also used this argument to show that the local search
algorithm for UFL was 3-approximation; their techniques have since been applied
to other facility location problems.
  In this paper, we give a proof of the $k$-median result which avoids this
coupling argument. These arguments can be used in other settings where the Arya
et al. arguments have been used. We also show that for the problem of opening
$k$ facilities $F$ to minimize the objective function $\Phi_p(F) = \big(\sum_{j
\in V} d(j, F)^p\big)^{1/p}$, the natural swap-based local-search algorithm is
a $\Theta(p)$-approximation. This implies constant-factor approximations for
$k$-medians (when $p=1$), and $k$-means (when $p = 2$), and an $O(\log
n)$-approximation algorithm for the $k$-center problem (which is essentially $p
= \log n$).
