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In this letter, we study a linear beamforming technique for weighted sum-rate (WSR) maximization in multiple-input single-output interfering broadcast channels. We focus on a distributed beamforming scheme which utilizes local channel state information (CSI) to mitigate inter-cell interference. In order to decouple the WSR maximization problem which involves the beamforming vectors of all base stations (BSs) into a distributed WSR problem as a function of local CSI, we apply high signal-to-interference-plus-noise ratio approximation. After defining the distributed WSR function, we solve the decoupled problems by using a zero-gradient based algorithm which converges to a local optimal point. Unlike conventional distributed schemes where additional information should be exchanged at each iteration, each BS of the proposed scheme utilizes only the local CSI to compute its beamforming vectors. Also we prove the convergence of the proposed algorithm. Simulation results show that our proposed algorithm exhibits the WSR performance almost identical to the centralized scheme with substantially reduced overhead.