Geometric Representation of Graphs in Low Dimension

Abstract

AbstractAn 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 boxicitybox(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, operation 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 has logn approximation ratio for boxicity 2 graphs. 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 1.5 (Δ+ 2) ln n dimensions, where Δ is the maximum degree of G. We also show that box(G) ≤ (Δ + 2) ln n for any graph G. Our bound is tight up to a factor of ln n. The only previously known general upper bound for boxicity was given by Roberts, namely box(G) ≤ n/2. Our result gives an exponentially better upper bound for bounded degree graphs.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, its boxicity is upper bound by c (d av + 1) ln n where d av is the average degree and c is a small constant. Also, we show that for any graph G, \(\ensuremath{\mathrm{box}}(G) \le \sqrt{8 n d_{av} \ln n}\), which is tight up to a factor of \(b \sqrt{\ln n}\) for a constant b.KeywordsPolynomial TimeMaximum DegreeIntersection GraphInterval GraphUnit Disk GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.