We formulate the problem of deciding which preference domains admit a non-dictatorial Arrovian Social Welfare Function as one of verifying the feasibility of an integer linear program. Many of the known results about the presence or absence of Arrovian Social Welfare Functions, impossibility theorems in Social Choice theory, and properties of majority rules etc., can be derived in a simple and unified way from this integer program. We also characterize those preference domains that admit a non-dictatorial, neutral Arrovian Soical Welfare function and a polyhedral characterization of Arrovian Social Welfare Functions on single-peaked domains.