Centrality of Trees for Capacitated k-Center

There is a large discrepancy in our understanding of uncapacitated and capacitated versions of network location problems. This is perhaps best illustrated by the classical k-center problem: there is a simple tight 2-approximation algorithm for the uncapacitated version whereas the first constant factor approximation algorithm for the general version with capacities was only recently obtained by using an intricate rounding algorithm that achieves an approximation guarantee in the hundreds. Our paper aims to bridge this discrepancy. For the capacitated k-center problem, we give a simple algorithm with a clean analysis that allows us to prove an approximation guarantee of 9. It uses the standard LP relaxation and comes close to settling the integrality gap (after necessary preprocessing), which is narrowed down to either 7, 8 or 9. The algorithm proceeds by first reducing to special tree instances, and then solves such instances optimally. Our concept of tree instances is quite versatile, and applies to natural variants of the capacitated k-center problem for which we also obtain improved algorithms. Finally, we give evidence to show that more powerful preprocessing could lead to better algorithms, by giving an approximation algorithm that beats the integrality gap for instances where all non-zero capacities are uniform.

💡 Research Summary

The paper addresses the capacitated k‑center problem, where one must select k facilities (centers) in a metric space and assign every client to an open facility while respecting individual facility capacities, with the objective of minimizing the maximum client‑to‑facility distance. While the uncapacitated version admits a simple 2‑approximation, the capacitated version has resisted comparable results; the first constant‑factor approximation appeared only recently (Cygan, Ha​jiaghayi, and Khuller, 2014) with a guarantee in the hundreds, and the integrality gap of the natural LP relaxation remained unbounded.

The authors present a conceptually simple algorithm that achieves a 9‑approximation, narrowing the gap between the algorithmic guarantee and the (post‑preprocessing) integrality gap, which they show must be either 7, 8, or 9. The approach consists of three main stages:

1. Guessing a lower bound τ on OPT – For a guessed value τ, they construct an unweighted graph G≤τ whose edges connect pairs of vertices at distance at most τ. Feasibility of a solution of value τ is equivalent to assigning each vertex to a neighbor in G≤τ.

2. Preprocessing using the Cygan‑et‑al. technique – The graph G≤τ may consist of several connected components. For each component Gi, they compute the smallest integer ki for which the standard LP relaxation LPki(Gi) is feasible. If the sum of these ki exceeds k, τ is certified to be smaller than OPT. The smallest τ for which this certification fails is denoted τ*. By construction, τ* ≤ OPT, and each component of G≤τ* admits a feasible LP solution.

3. Rounding the LP solution via tree reduction – Given a connected component G where the LP is feasible, the algorithm extracts a fractional solution (x, y) (xuv are assignment variables, yu are opening variables). The key technical contribution is the notion of distance‑r transfers: fractional opening mass can be moved only to vertices within a constant distance r, preserving the total opening mass and ensuring that local capacity does not decrease. To implement such transfers systematically, the authors reduce the rounding problem to a special class of instances they call tree instances.

   A tree instance is defined on a rooted tree where every internal (non‑leaf) node has opening variable y = 1, while leaves may have fractional openings. The authors devise a recursive algorithm that processes the tree from leaves upward. For each leaf, its fractional opening is pushed upward along the unique path to the root, possibly splitting the mass among several ancestors. When an internal node’s opening is fully transferred away, the node can be “closed” (its y set to 0) provided its capacity can be satisfied by nearby open nodes. The algorithm guarantees that each transfer occurs over a single edge (distance 1) and that the capacity of any node never drops below the demand of its incident clients. Because the original graph distances are at most τ* and each transfer adds at most a constant factor, the final assignment distance is bounded by 9·τ*.

The tree‑rounding algorithm is proved optimal for tree instances, which implies that the overall algorithm achieves a 9‑approximation for the original capacitated k‑center problem. Moreover, the authors explore a refined preprocessing step for the special case where capacities are either 0 or a uniform value L. By applying additional preprocessing and a more sophisticated rounding, they obtain a 6‑approximation for this uniform‑capacity scenario, thereby surpassing the basic integrality gap lower bound of 7.

The paper also discusses extensions to related problems, such as the capacitated k‑supplier problem and the budgeted opening problem with uniform capacities, showing that the same tree‑based rounding ideas yield improved guarantees.

In the concluding section, the authors outline future research directions. They conjecture that stronger preprocessing could potentially reduce the approximation factor to 3, matching the known hardness of approximation. They raise the question of whether lift‑and‑project hierarchies or alternative LP formulations might automatically capture the needed preprocessing. Finally, they suggest that the tree‑instance framework could be adapted to other capacitated location problems, including capacitated facility location and capacitated k‑median, where constant‑factor approximations remain open.

Overall, the work delivers a clean, LP‑based 9‑approximation algorithm for capacitated k‑center, substantially improving upon prior results, and introduces a versatile tree‑instance methodology that may have broader applicability in combinatorial optimization.