The $k$-Facility Location problem is a generalization of the classical problems $k$-Median and Facility Location. The goal is to select a subset of at most $k$ facilities that minimizes the total cost of opened facilities and established connections
between clients and opened facilities. We consider the hard-capacitated version of the problem, where a single facility may only serve a limited number of clients and creating multiple copies of a facility is not allowed. We construct approximation algorithms slightly violating the capacities based on rounding a fractional solution to the standard LP. It is well known that the standard LP (even in the case of uniform capacities and opening costs) has unbounded integrality gap if we only allow violating capacities by a factor smaller than $2$, or if we only allow violating the number of facilities by a factor smaller than $2$. In this paper, we present the first constant-factor approximation algorithms for the hard-capacitated variants of the problem. For uniform capacities, we obtain a $(2+varepsilon)$-capacity violating algorithm with approximation ratio $O(1/varepsilon^2)$; our result has not yet been improved. Then, for non-uniform capacities, we consider the case of $k$-Median, which is equivalent to $k$-Facility Location with uniform opening cost of the facilities. Here, we obtain a $(3+varepsilon)$-capacity violating algorithm with approximation ratio $O(1/varepsilon)$.
Given a set of $n$ terminals, which are points in $d$-dimensional Euclidean space, the minimum Manhattan network problem (MMN) asks for a minimum-length rectilinear network that connects each pair of terminals by a Manhattan path, that is, a path con
sisting of axis-parallel segments whose total length equals the pairs Manhattan distance. Even for $d=2$, the problem is NP-hard, but constant-factor approximations are known. For $d ge 3$, the problem is APX-hard; it is known to admit, for any $eps > 0$, an $O(n^eps)$-approximation. In the generalized minimum Manhattan network problem (GMMN), we are given a set $R$ of $n$ terminal pairs, and the goal is to find a minimum-length rectilinear network such that each pair in $R$ is connected by a Manhattan path. GMMN is a generalization of both MMN and the well-known rectilinear Steiner arborescence problem (RSA). So far, only special cases of GMMN have been considered. We present an $O(log^{d+1} n)$-approximation algorithm for GMMN (and, hence, MMN) in $d ge 2$ dimensions and an $O(log n)$-approximation algorithm for 2D. We show that an existing $O(log n)$-approximation algorithm for RSA in 2D generalizes easily to $d>2$ dimensions.
