  The Set-Union Knapsack Problem (SUKP) and Budgeted Maximum Coverage Problem (BMCP) are two closely related variant problems of the popular knapsack problem. Given a set of weighted elements and a set of items with nonnegative values, where each item covers several distinct elements, these two problems both aim to find a subset of items that maximizes an objective function while satisfying a knapsack capacity (budget) constraint. We propose an efficient and effective local search algorithm called E2LS for these two problems. To our knowledge, this is the first time that an algorithm has been proposed for both of them. E2LS trade-offs the search region and search efficiency by applying a proposed novel operator ADD$^*$ to traverse the refined search region. Such a trade-off mechanism allows E2LS to explore the solution space widely and quickly. The tabu search method is also applied in E2LS to help the algorithm escape from local optima. Extensive experiments on a total of 168 public instances with various scales demonstrate the excellent performance of the proposed algorithm for both the SUKP and BMCP. 
Efficient and Effective Local Search for the Set-Union Knapsack Problem   and Budgeted Maximum Coverage Problem