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 n1−ϵ-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=RP (the
weakest possible assumption).
(2) A tight n1/2−ϵ 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]