Geometric Representation of Graphs in Low Dimension.

An axis-parallel k-dimensional box is a Cartesian product R1 ×R2 ×⋯×Rk where Ri (for 1 ≤i ≤k) is a closed interval of the form [ai, bi] on the real line. For a graph G, its boxicity$\ensuremath{\mathrm{box}}(G)$ 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, 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 $\ensuremath{\mathrm{box}}(G) \le (\Delta + 2) \ln n$ 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 $\ensuremath{\mathrm{box}}(G) \le n/2$. 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·(dav + 1) ln n where dav 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. [ABSTRACT FROM AUTHOR]