In the Geometric Conflict Free Covering, we are given a set of points P, a set of closed geometric objects $\mathcal{O}$ and a conflict graph ${\mathcal{CG}}_{\mathcal{O}}$ with $\mathcal{O}$ as vertex set. An edge $(O_i,O_j)$ in ${\mathcal{CG}}_{\mathcal{O}}$ denotes conflict between $O_i$ and $O_j$ and at most one among these can be part of any feasible solution. A set of objects is conflict free if they form an independent set in ${\mathcal{CG}}_{\mathcal{O}}$. The objective is to find a conflict free set of objects that maximizes the number of points covered.

We consider the Unit Interval Covering where P is a set of points on the real line, and $\mathcal{O}$ is a set of closed unit-length intervals. The objective is to find a smallest subset of given intervals that covers P. We prove that for an arbitrary conflict graph the problem is Poly-APX -hard. We present an approximation algorithm for a special class of conflict graphs with a bounded graph parameter Clique Partition. A Clique Partition of the graph G is a set of cliques such that every vertex in the graph is part of exactly one clique. For any Clique Partition $\mathcal{C}$, we define the Clique Partition Graph, $G^{\mathcal{C}}$ with vertex set $\mathcal{C}$ and there is an edge $(C_i,C_j)$ in $G^{\mathcal{C}}$, if and only if there exist two vertices in G, $v_a\in C_i$ and $v_b\in C_j$ such that there is an edge $(v_a,v_b)$ in G. For a graph G, Clique Partition Chromatic Number is defined as the minimum chromatic number among all possible Clique Partitions of the Clique Partition Graph. In this paper, we consider those graph classes for which Clique Partition Chromatic Number can be computed in polynomial time.

We present a 4γ approximation algorithm for conflict graphs having Clique Partition Chromatic Number γ. We show that unit interval graphs and unit disk graphs have constant Clique Partition Chromatic Number while for chordal graphs, it is bounded by $\log n$. Note that, Clique Partition Chromatic Number is less than or equal to the chromatic number. Thus our algorithm achieves a constant approximation factor for graphs with constant chromatic number (e.g. planar graphs ). This is the first result regarding the approximability in Geometric Conflict Free Covering.

在无几何冲突覆盖中，我们得到一组点P，一组封闭的几何对象$\mathcal{Ø}$ 和一个冲突图 ${\mathcal{CG}}_{\mathcal{Ø}}$ 与 $\mathcal{Ø}$作为顶点集。优势$（{Ø}_{\mathrm{一世}}，{Ø}_{Ĵ}）$ 在 ${\mathcal{CG}}_{\mathcal{Ø}}$表示之间存在冲突${Ø}_{\mathrm{一世}}$ 和 ${Ø}_{Ĵ}$其中最多只能是任何可行解决方案的一部分。如果一组对象形成一个独立的集合，则它们是无冲突的${\mathcal{CG}}_{\mathcal{Ø}}$。目的是找到一个无冲突的对象集，该对象集可最大化覆盖的点数。

我们考虑单位间隔覆盖，其中P是实线上的一组点，并且$\mathcal{Ø}$是一组封闭的单位长度间隔。我们的目标是找出给定的间隔，涵盖的最小的子集P。我们证明对于任意冲突图，问题都是Poly-APX-hard。我们提出了一种带有边界图参数Clique Partition的特殊类冲突图的近似算法。图G的集团划分是集团的集合，使得图中的每个顶点都是一个集团的一部分。对于任何集团分区$\mathcal{C}$，我们定义了Clique分区图， $G^{\mathcal{C}}$ 带有顶点集 $\mathcal{C}$ 而且有优势 $（C_{\mathrm{一世}}，C_{Ĵ}）$ 在 $G^{\mathcal{C}}$，且仅当G中存在两个顶点时，$v_{\mathrm{一种}}\in C_{\mathrm{一世}}$ 和 $v_b\in C_{Ĵ}$ 这样有优势 $（v_{\mathrm{一种}}，v_b）$在G中。对于图G，“克利夫分区色度数”定义为“克利夫分区图”的所有可能克利夫分区中的最小色数。在本文中，我们考虑了可以在多项式时间内计算出Clique分区色数的图类。

对于具有Clique分区色度数γ的冲突图，我们提出了一种4γ近似算法。我们表明单位间隔图和单位圆盘图具有恒定的Clique分区色度数，而对于弦图，它由$\mathrm{日志}ñ$。请注意，“ Clique分区色度数”小于或等于色度数。因此，对于具有恒定色数的图形（例如，平面图），我们的算法获得了恒定的近似因子。这是有关无几何冲突覆盖的近似性的第一个结果。