Systems and methods for optimizing a query, and more particularly, systems and methods for finding optimal plans for graph queries by casting the task of finding the optimal plan as an integer programming (ILP) problem. A method for optimizing a query, comprises building a data structure for a query, the data structure including a plurality of components, wherein each of the plurality of components corresponds to at least one graph pattern, determining a plurality of flows of query variables between the plurality of components, and determining a combination of the plurality of flows between the plurality of components that results in a minimum cost to execute the query.