Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes |
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Authors: | L Sunil Chandran Mathew C Francis Naveen Sivadasan |
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Affiliation: | (3) Department of Industrial & Systems Engineering, Texas A&M University College Station, Texas, USA; |
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Abstract: | 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 $\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1}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
?1+\frac1clogn?d-1\lfloor 1+\frac{1}{c}\log n\rfloor^{d-1}
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. |
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Keywords: | |
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