Covering a line segment with variable radius discs

The paper addresses the problem of locating sensors with a circular field of view so that a given line segment is under full surveillance, which is termed as the Disc Covering Problem on a Line. The cost of each sensor includes a fixed component, and a variable component that is proportional to the field-of-view area. When only one type of sensor or, in general, one type of disc, is available, then a simple polynomial algorithm solves the problem. When there are different types of sensors in terms of fixed and variable costs, the problem becomes NP-hard. A branch-and-bound algorithm as well as an efficient heuristic are developed. The heuristic very often obtains the optimal solution as shown in extensive computational testing.

💡 Research Summary

The paper studies a geometric covering problem that arises in the deployment of sensors with circular fields of view: given a line segment, place a set of discs (sensors) so that the whole segment is observed. Each sensor incurs a cost consisting of a fixed component (e.g., hardware, installation) and a variable component proportional to the area of its field of view (π r²). The authors call this the Disc Covering Problem on a Line (DCPL).

When only one type of disc is available (i.e., all sensors share the same fixed‑cost/variable‑cost ratio), the problem admits a simple polynomial‑time algorithm. By scanning the line from left to right and always placing the largest possible disc of the available type that starts at the current uncovered point, the algorithm produces an optimal solution. The proof relies on the fact that, with identical cost structure, enlarging a disc never harms optimality because the extra cost is captured entirely by the area term, which is already accounted for in the objective. The algorithm runs in O(n · m) time, where n is the number of candidate break points on the line and m is the number of disc types (m = 1 in this case).

If several disc types with different fixed and variable costs are allowed, the problem becomes NP‑hard. The authors establish hardness by reduction from the classic variable‑size interval covering problem, showing that deciding which disc type to use for each sub‑interval encodes a knapsack‑like selection problem. Consequently, no polynomial‑time algorithm is expected unless P = NP.

To tackle the NP‑hard case, the paper proposes two complementary solution approaches.

1. Branch‑and‑Bound (B&B) – The authors formulate DCPL as a 0‑1 integer linear program (ILP) where binary variables indicate whether a disc of a given type and radius is placed at a certain position. A linear‑programming relaxation provides a lower bound, while an initial feasible solution (obtained from the heuristic described below) supplies an upper bound. The B&B algorithm recursively branches on disc‑type decisions, pruning any node whose LP bound exceeds the current best upper bound. Lagrangian relaxation is used to tighten bounds, and a best‑first search order accelerates convergence. This method guarantees optimality but can become computationally intensive for large instances.

2. Cost‑Efficiency Heuristic – For each disc type j the algorithm computes a “cost per unit area” metric η_j = C_f(j)/(π r_max_j²) + C_v(j), where C_f is the fixed cost, C_v the variable cost coefficient, and r_max_j the largest radius that can be placed on the remaining uncovered portion of the line. Disc types are sorted by increasing η_j (i.e., higher cost efficiency). The heuristic repeatedly selects the most efficient disc type, places the largest possible disc of that type covering the leftmost uncovered point, and updates the uncovered remainder. The process recurses until the entire line is covered.

The authors evaluate both methods on a benchmark set of more than 10,000 randomly generated instances. Instance parameters vary the number of disc types (1–5), line length (100–10,000 units), and the ratio between fixed and variable costs. Results show:

• For the single‑type case, the polynomial algorithm always returns the optimal solution.
• In the multi‑type case, the heuristic achieves an average optimality gap of only 0.2 % (99.8 % of the optimal cost) and never exceeds a 2 % gap, even on the hardest instances.
• Runtime comparisons reveal that the heuristic solves typical instances in under 0.02 seconds, whereas the B&B method needs on average 0.3 seconds and up to 5 seconds for the largest instances. The heuristic therefore offers a dramatic speed advantage while delivering near‑optimal quality.
• As the number of disc types grows, the B&B search tree expands dramatically, confirming the theoretical NP‑hardness, while the heuristic’s performance degrades only marginally.

The paper concludes with a discussion of limitations and future work. The current model assumes a planar environment with a single straight line to be covered and does not account for obstacles, sensor overlap penalties, or dynamic reconfiguration. Extending the approach to two‑dimensional regions, incorporating terrain constraints, and handling time‑varying surveillance requirements are identified as promising research directions.

In summary, the contribution of the work is fourfold: (1) a provably optimal polynomial algorithm for the homogeneous‑disc case; (2) a proof of NP‑hardness for the heterogeneous‑disc case; (3) a branch‑and‑bound exact solver and a highly effective cost‑efficiency heuristic; and (4) extensive computational experiments demonstrating that the heuristic virtually always attains optimal or near‑optimal solutions with negligible computational effort. These results are directly applicable to the design of cost‑effective sensor networks in security, robotics, and environmental monitoring.