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This paper addresses the minimum transmission broadcast problem in wireless networks and presents efficient solutions, including an optimal broadcast scheme and a distributed game-based algorithm. Distinct from related work in the literature which typically assumes wireless links are reliable, we address the issue of broadcasting over both reliable wireless links and unreliable wireless links. Our main contributions are as follows: We first formulate the minimum transmission broadcast problems over reliable links and over unreliable links as two mixed integer linear programming (MILP) problems, respectively. This way, optimal broadcast schemes can be easily obtained using any existing MILP solver, for small-scale networks. For large-scale networks, we propose a distributed game-based algorithm and prove that the game-based algorithm achieves Nash Equilibrium. Using simulation, we confirm that compared with existing algorithms in the literature and optimal solutions obtained by our MILP techniques, the proposed game-based algorithm performs very well in terms of delivery ratio, the number of transmissions, and convergence speed.