  Location Routing is a fundamental planning problem in logistics, in which
strategic location decisions on the placement of facilities (depots,
distribution centers, warehouses etc.) are taken based on accurate estimates of
operational routing costs. We present an approximation algorithm, i.e., an
algorithm with proven worst-case guarantees both in terms of running time and
solution quality, for the general capacitated version of this problem, in which
both vehicles and facilities are capacitated. Before, such algorithms were only
known for the special case where facilities are uncapacitated or where their
capacities can be extended arbitrarily at linear cost. Previously established
lower bounds that are known to approximate the optimal solution value well in
the uncapacitated case can be off by an arbitrary factor in the general case.
We show that this issue can be overcome by a bifactor approximation algorithm
that may slightly exceed facility capacities by an adjustable, arbitrarily
small margin while approximating the optimal cost by a constant factor. In
addition to these proven worst-case guarantees, we also assess the practical
performance of our algorithm in a comprehensive computational study, showing
that the approach allows efficient computation of near-optimal solutions for
instance sizes beyond the reach of current state-of-the-art heuristics.
