Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs

The study of graph products is a major research topic and typically concerns the term $f(G*H)$, e.g., to show that $f(G*H)=f(G)f(H)$. In this paper, we study graph products in a non-standard form $f(R[G*H]$ 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 $n^{1-\epsilon}$-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 $NP\neq RP$ (the weakest possible assumption). (2) A tight $n^{1/2-\epsilon}$ 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].