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# Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

Chandran, L Sunil and Francis, Mathew C and Sivadasan, Naveen (2010) Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes. In: Algorithmica, 56 (2). pp. 129-140.

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## Abstract

An axis-parallel k-dimensional box is a Cartesian product R-1 x R-2 x...x R-k where R-i (for 1 &lt;= i &lt;= 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 left perpendicular1 + 1/c log n right perpendicular(d-1) approximation ratio for any constant c &gt;= 1 when d &gt;= 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 left perpendicular(Delta + 2) ln nright perpendicular dimensions, where Delta is the maximum degree of G. This algorithm implies that box(G) &lt;= left perpendicular(Delta + 2) ln nright perpendicular 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 Delta, 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.

Item Type: Journal Article Copyright for this article belongs to Springer. Boxicity; Randomized algorithm; Derandomization; Random graph; Intersection graphs Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation) 19 Jan 2010 09:04 19 Sep 2010 05:54 http://eprints.iisc.ernet.in/id/eprint/25439

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