Interval representations of planar graphs
Journal of Combinatorial Theory Series B
Grid intersection graphs and boxicity
Discrete Mathematics - Special issue on combinatorics and algorithms
A special planar satisfiability problem and a consequence of its NP-completeness
Discrete Applied Mathematics
Label placement by maximum independent set in rectangles
WADS '97 Selected papers presented at the international workshop on Algorithms and data structure
Efficient approximation algorithms for tiling and packing problems with rectangles
Journal of Algorithms
Polynomial-Time Approximation Schemes for Geometric Intersection Graphs
SIAM Journal on Computing
Geometric representation of graphs in low dimension
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
Approximation algorithms for unit disk graphs
WG'05 Proceedings of the 31st international conference on Graph-Theoretic Concepts in Computer Science
Geometric representation of graphs in low dimension
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
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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 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(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.