Large genomic datasets are now created through numerous activities, including recreational genealogical investigations, biomedical research, and clinical care. At the same time, genomic data has become valuable for reuse beyond their initial point of collection, but privacy concerns often hinder access. Over the past several years, Beacon services have emerged to broaden accessibility to such data. These services enable users to query for the presence of a particular minor allele in a private dataset, information that can help care providers determine if genomic variation is spurious or has some known clinical indication. However, various studies have shown that even this limited access model can leak if individuals are members in the underlying dataset. Several approaches for mitigating this vulnerability have been proposed, but they are limited in that they 1) typically rely on heuristics and 2) offer probabilistic privacy guarantees, but neglect utility. In this paper, we present a novel algorithmic framework to ensure privacy in a Beacon service setting with a minimal number of query response flips (e.g., changing a positive response to a negative). Specifically, we represent this problem as combinatorial optimization in both the batch setting (where queries arrive all at once), as well as the online setting (where queries arrive sequentially). The former setting has been the primary focus in prior literature, whereas real Beacons allow sequential queries, motivating the latter investigation. We present principled algorithms in this framework with both privacy and, in some cases, worst-case utility guarantees. Moreover, through an extensive experimental evaluation, we show that the proposed approaches significantly outperform the state of the art in terms of privacy and utility.