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

TL;DR

This paper investigates complex graph product properties and establishes tight approximation hardness results for multiple problems in graph theory and computer science, revealing new subadditivity behaviors for certain graph parameters.

Contribution

It introduces a novel non-standard form of graph product analysis, demonstrating subadditivity and near subadditivity of key graph properties, leading to tight hardness of approximation results.

Findings

Subadditivity of induced and semi-induced matching numbers for certain graph products

Almost subadditivity of poset dimension in the studied graph product context

Tight hardness of approximation for multiple graph and combinatorial problems

Abstract

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: $f((G\oplus H)*J)$ where $G$, $H$ and $J$ are graphs, $\oplus$ 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 $\oplus$ and *, it is subadditive in the sense that $f((G\oplus H)*J)\leq f(G*J)+f(H*J)$. Moreover, when $f$ is the poset dimension number, it is almost subadditive. As applications of this result (we only need $J=K_2$ 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.