Boxicity and Poset Dimension

Abstract

Abstract. Let G be a simple, undirected, finite graph with vertex set V (G) and edge set E(G). A kdimensional box is a Cartesian product of closed intervals [a1, b1] × [a2, b2] × · · · × [a k , b k ]. 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 ) = 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)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 2 ∆) 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(n 0.5−ǫ ) for any ǫ > 0, unless N P = ZP P .

Boxicity and Separation Dimension

“…This is a direct consequence of the nontrivial fact that boxicity(G) ∈ O ∆ log 2 ∆ for any graph G of maximum degree ∆ [1]. It is known that there exist graphs of maximum degree ∆ whose boxicity can be as high as c∆ log ∆ [1], where c is a small enough positive constant. Let G be one such graph.…”

“…Let N (n, k) denote the cardinality of a smallest family of permutations that is k-suitable for [n]. In 1972, Spencer [30] proved that log log n ≤ N (n, 3) ≤ N (n, k) ≤ k2 k log log n. He also showed that N (n, 3) < log log n + 1 2 log log log n + log( √ 2π) + o (1). Fishburn and Trotter, in 1992, defined the dimension of a hypergraph on the vertex set [n] to be the minimum size of a family F of permutations of [n] such that every edge of the hypergraph is an intersection of initial segments of F [21].…”

“…are available in literature. Adiga, Bhowmick, and Chandran showed that the boxicity of a graph with maximum degree ∆ is O(∆ log 2 ∆) [1]. Chandran and Sivadasan proved that boxicity of a graph with treewidth t is at most t + 2 [15].…”

“…It was shown by Adiga, Chandran and Mathew that the boxicity of a k-degenerate graph on n vertices is O(k log n) [3]. Boxicity is also studied in relation with other dimensional parameters of graphs like partial order dimension and threshold dimension [1,32]. Studies on box representations of special graph classes too are available in abundance.…”

Sublinear approximation algorithms for boxicity and related problems

“…For any graph G on n vertices, box(G) ≤ n 2 and cub(G) ≤ 2n 3 . Upper bounds of boxicity in terms of parameters like maximum degree [5] and tree-width [6] are known. It was shown by Scheinerman [7] in 1984 that the boxicity of outer planar graphs is at most two.…”