Effective Variable Depth Local Search for the Budgeted Maximum Coverage Problem

Abstract

We address the Budgeted Maximum Coverage Problem (BMCP), which is a natural and more practical extension of the standard 0–1 knapsack problem and the set cover problem. Given m elements with nonnegative weights, n subsets of elements with nonnegative costs, and a total budget, BMCP aims to select some subsets such that the total cost of selected subsets does not exceed the budget, and the total weight of associated elements is maximized. In this paper, we propose a variable depth local search algorithm (VDLS) for the BMCP. VDLS first generates an initial solution by a greedy algorithm, then iteratively improves the solution through a partial depth-first search method, that can improve the solution by simultaneously changing the states (selected or not) of multiple subsets. Such method allows VDLS to explore the solution space widely and deeply, and to yield high-quality solutions. We further propose a neighbor structure to boost the algorithm performance, that is, both subsets have a neighbor relation if they share at least one common associated element. By applying the neighbor structure, VDLS can adjust the selected subsets while losing as few covered elements as possible. Since the existing BMCP benchmarks only have simple structures and small scales, we design 60 new instances with relatively large scales and complex structures to enrich the diversity of the BMCP instances. Experimental results on 30 public instances and 60 new instances we designed demonstrate that VDLS significantly outperforms the existing heuristic and the general CPLEX exact solver.

Efficient and Effective Local Search for the Set-Union Knapsack Problem and Budgeted Maximum Coverage Problem

“…For the BMCP, Li et al (Li et al, 2021) proposed the first local search method. Zhou et al (Zhou et al, 2022) proposed a local search algorithm based on a partial depth-first search tree.…”

“…The search region of the ATS-DLA algorithm (Zhou et al, 2021) is small, which may make the algorithm hard to escape from some local optima. The VDLS algorithm (Zhou et al, 2022) has a wide and deep search region. However, VDLS does not allow the current solution worse than the previous one, which may restrict the algorithm's search ability.…”

“…(Value-weight Ratio) The value-weight ratio of an item to a solution is defined as ( , ) = ∕ ( , ), which is the ratio of the value of to the addition weight of to . Obviously, an item with a larger value-weight ratio to a solution is a better candidate item of (Khuller et al, 1999;Zhou et al, 2021Zhou et al, , 2022.…”

“…The third category is the VDLS algorithm (Zhou et al, 2022), which is not based on tabu search but a partial depthfirst search method. VDLS has a large search region, and can yield significantly better solutions than the PLTS algorithm (Li et al, 2021).…”

“…• Set A: This set contains 30 instances with 585 to 1,000 items or elements. This set was proposed in (Li et al, 2021) and used in (Li et al, 2021;Zhou et al, 2022).…”