Fault-Tolerant Facility Location: a randomized dependent LP-rounding algorithm

We give a new randomized LP-rounding 1.725-approximation algorithm for the metric Fault-Tolerant Uncapacitated Facility Location problem. This improves on the previously best known 2.076-approximation algorithm of Swamy & Shmoys. To the best of our knowledge, our work provides the first application of a dependent-rounding technique in the domain of facility location. The analysis of our algorithm benefits from, and extends, methods developed for Uncapacitated Facility Location; it also helps uncover new properties of the dependent-rounding approach. An important concept that we develop is a novel, hierarchical clustering scheme. Typically, LP-rounding approximation algorithms for facility location problems are based on partitioning facilities into disjoint clusters and opening at least one facility in each cluster. We extend this approach and construct a laminar family of clusters, which then guides the rounding procedure. It allows to exploit properties of dependent rounding, and provides a quite tight analysis resulting in the improved approximation ratio.

💡 Research Summary

The paper addresses the Fault‑Tolerant Uncapacitated Facility Location (FTFL) problem, where each client j must be connected to r_j distinct facilities, and both facility opening costs f_i and connection costs c_{ij} obey the triangle inequality. Prior work achieved a 2.076‑approximation (Swamy & Shmoys) and the best known bound for the special case r_j=1 (UFL) is 1.5. The authors present a novel randomized algorithm that attains a 1.725‑approximation, the first to employ the dependent‑rounding technique in a facility‑location context.

The algorithm proceeds in several stages. First, the standard integer program is relaxed to a linear program, and an optimal fractional solution (x*, y*) is obtained. The solution is then uniformly scaled by a factor γ≈1.7245, after which each variable is truncated to 1, yielding a scaled fractional solution (ĥx, ĥy). Facilities with ĥy_i=1 are opened immediately, and the corresponding connections are fixed, incurring a cost increase of at most γ times the LP optimum.

The core technical contribution is a hierarchical, laminar clustering of facilities. For each non‑special client (clients with r_j>1 or without a “special” facility), two families of disjoint subsets, A_j and B_j, are maintained. Initially, A_j contains singleton sets for each “close” facility (the minimal set of nearest facilities that together provide at least r_j fractional service), while B_j is empty. The clustering algorithm repeatedly selects a client with the smallest maximal distance among its close facilities, extracts a minimal collection X_j of subsets from A_j whose fractional leftovers sum to at least the client’s residual demand, and merges them into a new cluster S_j. This new cluster is added to the laminar family, replaces the merged subsets in A_j, and updates the families of all other clients accordingly. The process continues until all residual demands are satisfied, after which a final cluster containing all facilities is added.

The laminar structure enables the use of dependent rounding in a controlled order: smaller clusters are rounded before larger ones that contain them. Dependent rounding guarantees three properties: (P1) marginal preservation, (P2) exact preservation of the total sum, and (P3) negative correlation. By applying it to each cluster in the laminar hierarchy, a refined property (P2′) holds for every cluster S: the number of opened facilities in S equals ⌊Σ_{i∈S} y_i⌋ with probability one. This deterministic lower bound on opened facilities per cluster is crucial for meeting the fault‑tolerance requirements.

A major theoretical advance is Theorem 2, which shows that for any subset S and integer k, the random variable Z = min{k, Σ_{i∈S} ŷ_i} obtained by dependent rounding has an expected value at least as large as the same quantity when each variable is rounded independently. Combined with a standard Chernoff‑type bound (Theorem 3), this yields the corollary that
E