Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

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 (a...

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Vydané v:	Algorithmica Ročník 56; číslo 2; s. 129 - 140
Hlavní autori:	Chandran, L. Sunil, Francis, Mathew C., Sivadasan, Naveen
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer-Verlag 01.02.2010 Springer
Predmet:	Algorithm Analysis and Problem Complexity Algorithmics. Computability. Computer arithmetics Algorithms Applied sciences Computer Science Computer science; control theory; systems Computer Systems Organization and Communication Networks Data Structures and Information Theory Exact sciences and technology Information retrieval. Graph Mathematics of Computing Theoretical computing Theory of Computation Boxicity Randomized algorithm Derandomization Intersection graphs Random graph Graph clique Graph theory Polynomial method Computational complexity Modeling Exact solution Polynomial time Upper bound NP hard problem Independent set Deterministic algorithms
ISSN:	0178-4617, 1432-0541
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Shrnutí:	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.
ISSN:	0178-4617 1432-0541
DOI:	10.1007/s00453-008-9163-5