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A region incrementing visual cryptography scheme (RIVCS) deals with the sharing of an image consisting of multiple regions with different secrecy levels, which can be incrementally revealed as the number of shares increases. The encoding basis matrices of RIVCS for an image containing three to five regions have been reported in the literature, but no construction method has ever been studied. We develop a novel and efficient construction for RIVCS using linear programming in this paper. The proposed integer linear program aims at the minimization of the pixel expansion under the constraints for being a RIVCS. Experimental results demonstrate the feasibility, applicability, and flexibility of our construction. The pixel expansions and contrasts derived from our scheme are also better than the previous results.