Graph Products Revisited: Tight Approximation Hardness of Induced Matching, Poset Dimension and More

Graph product is a fundamental tool with rich applications in both graph theory and theoretical computer science. It is usually studied in the form f(G∗H) where G and H are graphs, ∗ is a graph product and f is a graph property. For example, if f is the independence number and ∗ is the disjunctive product, then the product is known to be multiplicative: f(G∗H) = f(G)f(H). In this paper, we study graph products in the following non-standard form: f((G⊕H) ∗ J) where G, H and J are graphs, ⊕ and ∗ are two different graph products and f is a graph property. We show that if f is the induced and semi-induced matching number, then for some products ⊕ and ∗, it is subadditive in the sense that f((G ⊕ H) ∗ J) ≤ f(G ∗ J) + f(H ∗ J). Moreover, when f is the poset dimension number, it is almost subadditive. As applications of this result (we only need J = K2 here), we obtain tight hardness of approximation for various problems in discrete mathematics and computer science: bipartite induced and semi-induced matching (a.k.a. maximum expanding sequences), poset dimension, maximum feasible subsystem with 0/1 coefficients, unit-demand min-buying and single-minded pricing, donation center location, boxicity, cubicity, threshold dimension and independent packing.