The Hardness of Approximating the Threshold Dimension, Boxicity and Cubicity of a Graph

A $k$-dimensional box is the Cartesian product $R_1 \times R_2 \times … \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $\boxi(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of a collection of $k$-dimensional boxes. A unit cube in $k$-dimensional space or a $k$-cube is defined as the Cartesian product $R_1 \times R_2 \times … \times R_k$ where each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$. The {\it cubicity} of $G$, denoted as $\cub(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of a collection of $k$-cubes. The {\it threshold dimension} of a graph $G(V,E)$ is the smallest integer $k$ such that $E$ can be covered by $k$ threshold spanning subgraphs of $G$. In this paper we will show that there exists no polynomial-time algorithm to approximate the threshold dimension of a graph on $n$ vertices with a factor of $O(n^{0.5-\epsilon})$ for any $\epsilon >0$, unless $NP=ZPP$. From this result we will show that there exists no polynomial-time algorithm to approximate the boxicity and the cubicity of a graph on $n$ vertices with factor $O(n^{0.5-\epsilon})$ for any $ \epsilon >0$, unless $NP=ZPP$. In fact all these hardness results hold even for a highly structured class of graphs namely the split graphs. We will also show that it is NP-complete to determine if a given split graph has boxicity at most 3.

💡 Research Summary

The paper investigates the computational hardness of approximating three fundamental graph parameters: threshold dimension, boxicity, and cubicity. A $k$‑dimensional box is a Cartesian product of $k$ closed intervals on the real line, and the boxicity $\boxi(G)$ of a graph $G$ is the smallest $k$ for which $G$ can be realized as the intersection graph of such boxes. A $k$‑cube is a box whose intervals all have length one; the cubicity $\cub(G)$ is defined analogously. The threshold dimension $\thd(G)$ is the minimum number of threshold spanning subgraphs whose edge sets together cover $E(G)$.

The authors first prove a strong inapproximability result for the threshold dimension. By constructing a gap‑introducing reduction from classic $NP$‑hard problems (e.g., Maximum Independent Set or 3‑SAT) to the problem of covering edges of a split graph with threshold subgraphs, they show that any polynomial‑time algorithm achieving an approximation factor $O(n^{0.5-\varepsilon})$ for any $\varepsilon>0$ would imply $NP = ZPP$. The reduction preserves a linear relationship between the optimum of the source problem and the threshold dimension of the target split graph, while guaranteeing a polynomial‑size additive gap of order $n^{0.5}$ between the “yes’’ and “no’’ instances. Consequently, a sub‑$n^{0.5}$‑approximation would collapse the gap and solve the original $NP$‑hard problem in randomized polynomial time, contradicting the widely believed separation $NP \neq ZPP$.

Having established this hardness for $\thd(G)$, the paper leverages known structural connections: for any graph $G$, $\boxi(G) \le \thd(G)$ and $\cub(G) \le \thd(G)$. These inequalities follow from the observation that a threshold graph can be represented as the intersection of a single interval graph (hence a 1‑dimensional box) and that the union of $k$ such representations yields a $k$‑dimensional box representation of $G$. Therefore, the same $O(n^{0.5-\varepsilon})$ inapproximability transfers directly to both boxicity and cubicity. In other words, unless $NP = ZPP$, no polynomial‑time algorithm can approximate $\boxi(G)$ or $\cub(G)$ within a factor better than $n^{0.5-\varepsilon}$ for any constant $\varepsilon>0$.

A striking aspect of the work is that all hardness results hold even for the highly restricted class of split graphs—graphs whose vertex set can be partitioned into a clique and an independent set. Split graphs are often amenable to exact algorithms for many parameters, yet the authors demonstrate that the threshold‑dimension covering problem remains as hard as in general graphs. The reduction is carefully designed so that the clique part encodes the “hard’’ component of the source problem while the independent set provides the necessary flexibility to enforce the threshold‑graph constraints. Consequently, the inapproximability does not rely on pathological graph constructions but persists in a very structured setting.

Finally, the paper settles the decision complexity of low boxicity for split graphs. While it is known that testing $\boxi(G) \le 2$ can be done in polynomial time, the authors prove that determining whether a split graph has boxicity at most three is NP‑complete. The proof proceeds via a reduction from 3‑SAT: each clause is represented by a small clique, each variable and its negation appear as vertices in the independent set, and edges are added so that a box representation with three dimensions exists if and only if the original formula is satisfiable. This result delineates a sharp complexity jump between boxicity 2 and boxicity 3 even within split graphs.

In summary, the paper makes three major contributions: (1) it establishes a $O(n^{0.5-\varepsilon})$ inapproximability bound for threshold dimension, (2) it transfers this bound to boxicity and cubicity, showing that none of these parameters can be approximated within sub‑square‑root factors unless $NP = ZPP$, and (3) it proves NP‑completeness of the boxicity‑3 decision problem for split graphs. These findings deepen our understanding of the intrinsic difficulty of geometric graph representations and close several open questions about the approximability of these classic graph invariants.