Boxicity and Poset Dimension

Let G be a simple, undirected, finite graph with vertex set V (G) and edge set E(G). A k-dimensional box is a Cartesian product of closed intervals [a1, b1]× [a2, b2]× · · · × [ak, bk]. The boxicity of G, box(G) is the minimum integer k such that G can be represented as the intersection graph of k-dimensional boxes, i.e. each vertex is mapped to a k-dimensional box and two vertices are adjacent in G if and only if their corresponding boxes intersect. Let P = (S, P ) be a poset where S is the ground set and P is a reflexive, anti-symmetric and transitive binary relation on S. The dimension of P, dim(P) is the minimum integer t such that P can be expressed as the intersection of t total orders. Let GP be the underlying comparability graph of P, i.e. S is the vertex set and two vertices are adjacent if and only if they are comparable in P. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset P, box(GP )/(χ(GP ) − 1) ≤ dim(P) ≤ 2box(GP ), where χ(GP ) is the chromatic number of GP and χ(GP ) 6= 1. It immediately follows that if P is a height-2 poset, then box(GP ) ≤ dim(P) ≤ 2box(GP ) since the underlying comparability graph of a height-2 poset is a bipartite graph. The second result of the paper relates the boxicity of a graph G with a natural partial order associated with the extended double cover of G, denoted as Gc: Note that Gc is a bipartite graph with partite sets A and B which are copies of V (G) such that corresponding to every u ∈ V (G), there are two vertices uA ∈ A and uB ∈ B and {uA, vB} is an edge in Gc if and only if either u = v or u is adjacent to v in G. Let Pc be the natural height-2 poset associated with Gc by making A the set of minimal elements and B the set of maximal elements. We show that box(G) 2 ≤ dim(Pc) ≤ 2box(G) + 4. These results have some immediate and significant consequences. The upper bound dim(P) ≤ 2box(GP ) allows us to derive hitherto unknown upper bounds for poset dimension such as dim(P) ≤ 2 tree-width (GP ) + 4, since boxicity of any graph is known to be at most its tree-width + 2. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree ∆ is O(∆ log ∆) which is an improvement over the best known upper bound of ∆2 + 2. (2) There exist graphs with boxicity Ω(∆ log ∆). This disproves a conjecture that the boxicity of a graph is O(∆). (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on n vertices with a factor of O(n0.5− ) for any > 0, unless NP = ZPP .