In this paper, we study the Maximum Happy Vertices and the Maximum Happy Edges problems (MHV and MHE for short). Very recently, the problems attracted a lot of attention and were studied in Agrawal '18, Aravind et al. '16, Choudhari and Reddy '18, Misra and Reddy '18. Main focus of our work is lower bounds on the computational complexity of these problems. Established lower bounds can be divided into the following groups: NP-hardness of the above guarantee parameterization, kernelization lower bounds (answering questions of Misra and Reddy '18), exponential lower bounds under the Set Cover Conjecture and the Exponential Time Hypothesis, and inapproximability results. Moreover, we present an ${\mathcal{O}}^{⁎}({\ell }^k)$ randomized algorithm for MHV and an ${\mathcal{O}}^{⁎}(2^k)$ algorithm for MHE, where ℓ is the number of colors used and k is the number of required happy vertices or edges. These algorithms cannot be improved to subexponential taking proved lower bounds into account.

在本文中，我们研究了最大快乐顶点和最大快乐边缘问题（简称MHV和MHE）。最近，这些问题引起了很多关注，并在Agrawal '18，Aravind等人的文章中进行了研究。'16，乔达里和雷迪'18，米斯拉和雷迪'18。我们工作的主要重点是这些问题的计算复杂度的下限。确定的下界可以分为以下几类：上面保证参数化的NP硬度，内核化下界（回答Misra和Reddy '18的问题），“ Set Cover Conjecture”和“指数时间假说”下的指数下界以及不可逼近性结果。此外，我们提出了一个${\mathcal{Ø}}^{⁎}（{\ell }^{ķ}）$ MHV的随机算法和 ${\mathcal{Ø}}^{⁎}（2^{ķ}）$MHE的算法，其中ℓ是使用的颜色数，k是所需的快乐顶点或边数。考虑到已证明的下界，无法将这些算法改进为次指数。