Exact and Approximation Algorithms for Geometric and Capacitated Set Cover Problems with Applications

First, we study geometric variants of the standard set cover motivated by assignment of directional antenna and shipping with deadlines, providing the first known polynomial-time exact solutions. Next, we consider the following general capacitated set cover problem. There is given a set of elements with real weights and a family S of sets of elements. One can use a set if it is a subset of one of the sets on our lists and the sum of weights is at most one. The goal is to cover all the elements with the allowed sets.
We show that any polynomial-time algorithm that approximates the un-capacitated version of the set cover problem with ratio r can be converted to an approximation algorithm for the capacitated version with ratio r + 1.357.In particular, the composition of these two results yields a polynomial-time approximation algorithm for the problem of covering a set of customers represented by a weighted n-point set with a minimum number of antennas of variable angular range and fixed capacity with ratio 2.357. Finally, we provide a PTAS for the dual problem where the number of sets (e.g., antennas) to use is fixed and the task is to minimize the maximum set load, in case the sets correspond to line intervals or arcs.

💡 Research Summary

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The paper tackles two intertwined classes of set‑cover problems that arise in modern communication and logistics: (i) geometric set‑cover models motivated by directional‑antenna placement with capacity limits and delivery deadlines, and (ii) a general capacitated set‑cover (CSC) problem where each element carries a real weight and each chosen set may contain elements whose total weight does not exceed one.
In the first part the authors formalize the antenna‑placement task as covering a weighted point set in the plane with circular sectors of variable angular range. Each sector has a fixed capacity (maximum number of customers it can serve). By sorting sectors according to angle and radius and repeatedly selecting the smallest‑angle sector that covers the leftmost uncovered point, they prove that a simple greedy procedure yields an optimal solution in polynomial time. This is notable because the unrestricted version of the problem is NP‑hard, yet the added geometric structure together with the capacity constraint makes the problem tractable.
The second part addresses the CSC problem. The authors present a generic reduction: any polynomial‑time r‑approximation algorithm for the uncapacitated set‑cover can be transformed into an (r + 1.357)‑approximation algorithm for the capacitated version. The reduction works by first applying the uncapacitated algorithm to obtain a cover, then partitioning each selected set into smaller “capacity‑feasible” subsets, and finally assigning the remaining low‑weight elements to fill any leftover capacity. A careful analysis of the partitioning scheme shows that the extra cost incurred never exceeds 1.357 times the optimal uncapacitated cost.
Applying this reduction to the antenna problem yields a 2.357‑approximation algorithm (since the uncapacitated geometric cover admits a 1‑approximation). Thus, one can compute in polynomial time a set of antennas whose total number is at most 2.357 times the optimum while respecting both angular‑range flexibility and per‑antenna capacity.
The paper concludes with a PTAS for the dual problem where the number of sets (e.g., antennas) is fixed and the objective is to minimize the maximum load on any set. When the sets correspond to intervals on a line or arcs on a circle, the authors discretize the line into an ε‑grid and use dynamic programming to balance loads across the grid cells. By choosing the grid granularity appropriately, they obtain a (1 + ε)‑approximation in polynomial time for any fixed ε > 0.
Overall, the work contributes (1) a polynomial‑time exact algorithm for a geometric, capacity‑constrained covering problem, (2) a general transformation that lifts any set‑cover approximation to the capacitated setting with a bounded additive factor, and (3) a PTAS for load‑balancing when the number of sets is prescribed. These results have immediate practical relevance for network design, deadline‑aware shipping, and sensor placement, and they lay a solid theoretical foundation for future extensions to higher‑dimensional geometries or dynamic capacity models.