We consider a well-studied generalization of the maximum clique problem which is defined as follows. Given a graph G on n vertices and a fixed parameter \(d\ge 1\), in the maximum diameter-bounded subgraph problem (MaxDBS for short) the goal is to find a (vertex) maximum subgraph of G of diameter at most d. For \(d=1\), this problem is equivalent to the maximum clique problem and thus it is NP-hard to approximate it within a factor \(n^{1-\epsilon }\), for any \(\epsilon >0\). Moreover, it is known that, for any \(d\ge 2\), it is NP-hard to approximate MaxDBS within a factor \(n^{1/2-\epsilon }\), for any \(\epsilon >0\). In this paper we focus on MaxDBS for the class of unit disk graphs. We provide a polynomial-time constant-factor approximation algorithm for the problem. The approximation ratio of our algorithm does not depend on the diameter d. Even though the algorithm itself is simple, its analysis is rather involved. We combine tools from the theory of hypergraphs with bounded VC-dimension, k-quasi planar graphs, fractional Helly theorems, and several geometric properties of unit disk graphs.

我们考虑对最大集团问题进行充分研究的概括，其定义如下。给定n个顶点上的图G和固定参数\(d\ge 1\)，在最大直径有界子图问题（简称 MaxDBS）中，目标是找到最大直径G的 （顶点）最大子图d。对于\（d = 1 \） ，这个问题是相当于最大团问题，因此它是NP-难以倍之内近似它\（N ^ {1- \小量} \） ，对于任何\（\小量>0\)。此外，众所周知，对于任何\(d\ge 2\)，在一个因子内近似 MaxDBS 是 NP 难的\(n^{1/2-\epsilon }\)，对于任何\(\epsilon >0\)。在本文中，我们专注于单位磁盘图类的 MaxDBS。我们为该问题提供了多项式时间常数因子近似算法。我们算法的近似比不取决于直径 d。尽管算法本身很简单，但它的分析却相当复杂。我们将超图理论中的工具与有界 VC 维、k拟平面图、分数 Helly 定理和单位圆盘图的几个几何性质相结合。