We address the Traveling Salesman Problem (TSP), a famous NP-hard combinatorial optimization problem. And we propose a variable strategy reinforced approach, denoted as VSR-LKH, which combines three reinforcement learning methods (Q-learning, Sarsa and Monte Carlo) with the well-known TSP algorithm, called Lin-Kernighan-Helsgaun (LKH). VSR-LKH replaces the inflexible traversal operation in LKH, and lets the program learn to make choice at each search step by reinforcement learning. Experimental results on 111 TSP benchmarks from the TSPLIB with up to 85,900 cities demonstrate the excellent performance of the proposed method.

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.

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