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We propose a new integer-linear programming model for the delete relaxation in cost-optimal planning. While a straightforward IP for the delete relaxation is impractical, our enhanced model incorporates variable reduction techniques based on landmarks, relevance-based constraints, dominated action elimination, immediate action application, and inverse action constraints, resulting in an IP that can be used to directly solve delete-free planning problems. We show that our IP model is competitive with previous state-of-the-art solvers for delete-free problems. The LP-relaxation of the IP model is often a very good approximation to the IP, providing an approach to approximating the optimal value of the delete-free task that is complementary to the well-known LM-cut heuristic. We also show that constraints that partially consider delete effects can be added to our IP/LP models. We embed the new IP/LP models into a forward-search based planner, and show that the performance of the resulting planner on standard IPC benchmarks is comparable with the state-of-the-art for cost-optimal planning.