Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes - Chandran, L. Sunil; Francis, Mathew C.; Sivadasan, Naveen (Algorithmica, 2010)
An axis-parallel k -dimensional box is a Cartesian product R 1 × R 2 × ⋅⋅⋅ × R k where R i (for 1≤ i ≤ k ) is a closed interval of the form [ a i , b i ] on the real line. For a graph G , its boxicity box ( G ) is the minimum dimension k , such that G is representable as the intersection graph of (axis-parallel) boxes in k -dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a approximation ratio for any constant c ≥1 when d ≥2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard. We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in ⌈(Δ+2)ln n ⌉ dimensions, where Δ is the maximum degree of G . This algorithm implies that box ( G )≤⌈(Δ+2)ln n ⌉ for any graph G . Our bound is tight up to a factor of ln n . We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree Δ, we show that for almost all graphs on n vertices, their boxicity is O ( d av ln n ) where d av is the average degree.