高维圆盘图最大团的逼近性和不可逼近性
Approximation and Inapproximability Results for Maximum Clique of Disc Graphs in High Dimensions
摘要：
我们证明了算法和硬度的问题，找到最大的一个固定的直径在欧几里得空间的结果。特别是，我们证明，如果$A^*$是最大的子集的直径$r$的$n$点在欧几里得空间，然后为每个$ε>0$存在一个多项式时间算法，输出一组$B$的大小至少$|A^*|$和直径最多$r(\sqrt{2}+ε)$。在硬度方面，粗略地说，我们表明，除非$P=NP$为每个$ε>0$它是不可能保证直径$r(\sqrt{4/3}-ε)$为$B$，即使算法被允许输出一组大小$({95\over 94}-ε)^{-1}|A^*|$。
英文摘要：
We prove algorithmic and hardness results for the problem of finding the largest set of a fixed diameter in the Euclidean space. In particular, we prove that if $A^*$ is the largest subset of diameter $r$ of $n$ points in the Euclidean space, then for every $ε>0$ there exists a polynomial time algorithm that outputs a set $B$ of size at least $|A^*|$ and of diameter at most $r(\sqrt{2}+ε)$. On the hardness side, roughly speaking, we show that unless $P=NP$ for every $ε>0$ it is not possible to guarantee the diameter $r(\sqrt{4/3}-ε)$ for $B$ even if the algorithm is allowed to output a set of size $({95\over 94}-ε)^{-1}|A^*|$.
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