We study the complexity of computing (and approximating) VC Dimension and Littlestone's Dimension when we are given the concept class explicitly. We give a simple reduction from Maximum (Unbalanced) Biclique problem to approximating VC Dimension and Littlestone's Dimension. With this connection, we derive a range of hardness of approximation results and running time lower bounds. For example, under the (randomized) Gap-Exponential Time Hypothesis or the Strongish Planted Clique Hypothesis, we show a tight inapproximability result: both dimensions are hard to approximate to within a factor of $o(\log n)$ in polynomial-time. These improve upon constant-factor inapproximability results from [Manurangsi and Rubinstein, COLT 2017].

翻译：当明确给予概念类时,我们研究计算(和接近)VC维度和Littestone维度的复杂程度。我们从最大(不平衡)二元问题到接近VC维度和Littestone维度的简单减少。与此相关,我们得出一系列近似结果的硬度和运行时间下限。例如,在(随机的)差异 - 分散时间假设或强力规划的克隆伪证下,我们显示出一种接近性很强的结果:两种维度很难在多米时的美元(log n)系数内接近。这些在常态-因素的不协调性结果从[Manurangsi 和Rubinstein,COLT 2017]中得到改善。