Probability Learning based Tabu Search for the Budgeted Maximum Coverage Problem

Knapsack problems are classic models that can formulate a wide range of applications. In this work, we deal with the Budgeted Maximum Coverage Problem (BMCP), which is a generalized 0-1 knapsack problem. Given a set of items with nonnegative weights and a set of elements with nonnegative profits, where each item is composed of a subset of elements, BMCP aims to pack a subset of items in a capacity-constrained knapsack such that the total weight of the selected items does not exceed the knapsack capacity, and the total profit of the associated elements is maximized. Note that each element is counted once even if it is covered multiple times. BMCP is closely related to the Set-Union Knapsack Problem (SUKP) that is well studied in recent years. As the counterpart problem of SUKP, however, BMCP was introduced early in 1999 but since then it has been rarely studied, especially there is no practical algorithm proposed. By combining the reinforcement learning technique to the local search procedure, we propose a probability learning based tabu search (PLTS) algorithm for addressing this NP-hard problem. The proposed algorithm iterates through two distinct phases, namely a tabu search phase and a probability learning based perturbation phase. As there is no benchmark instances proposed in the literature, we generate 30 benchmark instances with varied properties. Experimental results demonstrate that our PLTS algorithm significantly outperforms the general CPLEX solver for solving the challenging BMCP in terms of the solution quality.