圆弧图盒性的常数因子近似算法
A Constant Factor Approximation Algorithm for Boxicity of Circular Arc Graphs
摘要：
图$G(V,E)$的盒性是使$G$可以表示为$\mathbf{R}^k$中$k$维轴平行矩形的交图的最小整数$k$。等价地，它是顶点集$V$上使得它们的边集的交是$E$的区间图的最小数目。已知盒性不能被近似，即使对于像二分图、共二分图和分裂图这样的图类低于$O(n^{0.5 - ε})$ -因子，对于任何$ε>0$在多项式时间内，除非$NP=ZPP$。到目前为止，还没有一个众所周知的图类的无界boxicity，甚至一个$n^ε$因子近似算法计算boxicity是已知的，任何$ε<1$。本文研究了圆弧图--圆弧交图的盒性问题。我们给出了一个计算任意圆弧图盒度的$(2+\frac{1}{k})$ -因子多项式时间近似算法以及相应的盒表示，其中$k \ge 1$是它的盒度.对于正态圆弧（NCA）图，在给定NCA模型的情况下，该算法可以改进为一个加性2因子近似算法。在这两种情况下，近似计算盒性的算法的时间复杂度都是$O(mn+n^2)$，在$O(mn+kn^2)= O(n^3)$时间中，我们还得到了它们相应的盒表示，其中$n$是图的顶点数，$m$是其边数。加性2因子算法直接适用于任何真圆弧图，因为计算它的NCA模型可以在多项式时间内完成。
英文摘要：
Boxicity of a graph $G(V,E)$ is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional axis parallel rectangles in $\mathbf{R}^k$. Equivalently, it is the minimum number of interval graphs on the vertex set $V$ such that the intersection of their edge sets is $E$. It is known that boxicity cannot be approximated even for graph classes like bipartite, co-bipartite and split graphs below $O(n^{0.5 - ε})$-factor, for any $ε>0$ in polynomial time unless $NP=ZPP$. Till date, there is no well known graph class of unbounded boxicity for which even an $n^ε$-factor approximation algorithm for computing boxicity is known, for any $ε<1$. In this paper, we study the boxicity problem on Circular Arc graphs - intersection graphs of arcs of a circle. We give a $(2+\frac{1}{k})$-factor polynomial time approximation algorithm for computing the boxicity of any circular arc graph along with a corresponding box representation, where $k \ge 1$ is its boxicity. For Normal Circular Arc(NCA) graphs, with an NCA model given, this can be improved to an additive 2-factor approximation algorithm. The time complexity of the algorithms to approximately compute the boxicity is $O(mn+n^2)$ in both these cases and in $O(mn+kn^2)= O(n^3)$ time we also get their corresponding box representations, where $n$ is the number of vertices of the graph and $m$ is its number of edges. The additive 2-factor algorithm directly works for any Proper Circular Arc graph, since computing an NCA model for it can be done in polynomial time.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
