# The hardness of approximating the boxicity, cubicity and threshold dimension of a graph

Adiga, Abhijin and Bhowmick, Diptendu and Chandran, Sunil L (2010) The hardness of approximating the boxicity, cubicity and threshold dimension of a graph. In: Discrete Applied Mathematics, 158 (16). pp. 1719-1726.

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Official URL: http://dx.doi.org/10.1016/j.dam.2010.06.017

## Abstract

A k-dimensional box is the Cartesian product R-1 X R-2 X ... X R-k where each R-i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R-1 X R-2 X ... X R-k where each R-i is a closed interval oil the real line of the form a(i), a(i) + 1]. The cubicity of G, denoted as cub(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-cubes. The threshold dimension of a graph G(V, E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. In this paper we will show that there exists no polynomial-time algorithm for approximating the threshold dimension of a graph on n vertices with a factor of O(n(0.5-epsilon)) for any epsilon > 0 unless NP = ZPP. From this result we will show that there exists no polynomial-time algorithm for approximating the boxicity and the cubicity of a graph on n vertices with factor O(n(0.5-epsilon)) for any epsilon > 0 unless NP = ZPP. In fact all these hardness results hold even for a highly structured class of graphs, namely the split graphs. We will also show that it is NP-complete to determine whether a given split graph has boxicity at most 3. (C) 2010 Elsevier B.V. All rights reserved.

Item Type: Journal Article Discrete Applied Mathematics Elsevier Science Copyright of this article belongs to Elsevier Science. Boxicity;Cubicity;Threshold dimension;Partial order dimension;Split graph;NP-completeness;Approximation hardness. Division of Electrical Sciences > Computer Science & Automation 18 Oct 2010 07:26 29 Feb 2012 07:19 http://eprints.iisc.ac.in/id/eprint/33282