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With the extensive application of laser cutting, the proper planning of the cutting path has a significate impact on industrial production, like clothing, metalware, and furniture. This paper proposes heuristic approaches for the two-dimensional cutting path problem, which aims to find the shortest path to cut pieces from the master plate. Two cutting methods, the arc routing method, and the node routing method, are studied based on whether the piece must be cut separately. A two-step heuristic approach is proposed for the arc routing method. A model based on a generalized traveling salesman problem (GTSP) and a variable neighborhood search (VNS) is introduced for the node routing method. The results show that the proposed two-step heuristic quickly finds the optimal or near-optimal solution. CPLEX using the GTSP model can solve small-size instances, while the VNS can find the solution with good quality in a reasonable time for medium-size instances. In addition, the model and approaches are also tested on larger instances with more than 250 pieces and 20,000 nodes. The results show that the proposed heuristics can handle these instances within a reasonable time. • A two-step heuristic is proposed for the arc routing method. • A new model is proposed for the node routing method. • A variable neighborhood search is introduced for the node routing method. • The two-step heuristic quickly finds the optimal or near-optimal solution. [ABSTRACT FROM AUTHOR]