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The study of graph products is a major research topic and typically concerns the term f(GβH), e.g., to show that f(GβH)=f(G)f(H). In this paper, we study graph products in a non-standard form f(R[GβH] where R is a "reduction", a transformation of any graph into an instance of an intended optimization problem. We resolve some open problems as applications. (1) A tight n1βΟ΅-approximation hardness for the minimum consistent deterministic finite automaton (DFA) problem, where n is the sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this implies the hardness of properly learning DFAs assuming NPξ =RP (the weakest possible assumption). (2) A tight n1/2βΟ΅ hardness for the edge-disjoint paths (EDP) problem on directed acyclic graphs (DAGs), where n denotes the number of vertices. (3) A tight hardness of packing vertex-disjoint k-cycles for large k. (4) An alternative (and perhaps simpler) proof for the hardness of properly learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004 and J. Comput.Syst.Sci. 2008]
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