Recently, Ivan Mihajlin and Alexander Smal proved a composition theorem of a universal relation and some function via so called xor composition, that is there exists some function $f:\{0,1\}^n \rightarrow \{0,1\}$ such that $\textsf{CC}(\text{U}_n \diamond \text{KW}_f) \geq 1.5n-o(n)$ where $\textsf{CC}$ denotes the communication complexity of the problem. In this paper, we significantly improve their result and present an asymptotically tight and much more general composition theorem of a universal relation and most functions, that is for most functions $f:\{0,1\}^n \rightarrow \{0,1\}$ we have $\textsf{CC}(\text{U}_m \diamond \text{KW}_f) \geq m+ n -O(\sqrt{m})$ when $m=\omega(\log^2 n),n =\omega(\sqrt{m})$. This is done by a direct proof of composition theorem of a universal relation and a multiplexor in the partially half-duplex model avoiding the xor composition. And the proof works even when the multiplexor only contains a few functions. One crucial ingredient in our proof involves a combinatorial problem of constructing a tree of many leaves and every leaf contains a non-overlapping set of functions. For each leaf, there is a set of inputs such that every function in the leaf takes the same value, that is all functions are restricted. We show how to choose a set of good inputs to effectively restrict these functions to force that the number of functions in each leaf is as small as possible while maintaining the total number of functions in all leaves. This results in a large number of leaves.

翻译：最近，Ivan Mihajlin 和 Alexander Smal 通过所谓的异或组合证明了通用关系与某个函数的组合定理，即存在某个函数 $f:\{0,1\}^n \rightarrow \{0,1\}$，使得 $\textsf{CC}(\text{U}_n \diamond \text{KW}_f) \geq 1.5n-o(n)$，其中 $\textsf{CC}$ 表示问题的通信复杂度。在本文中，我们显著改进了他们的结果，并提出了一个渐近紧且更具一般性的通用关系与多数函数的组合定理，即对大多数函数 $f:\{0,1\}^n \rightarrow \{0,1\}$，当 $m=\omega(\log^2 n), n=\omega(\sqrt{m})$ 时，有 $\textsf{CC}(\text{U}_m \diamond \text{KW}_f) \geq m+n-O(\sqrt{m})$。这是通过在部分半双工模型中直接证明通用关系与多路复用器的组合定理（避免使用异或组合）实现的，且该证明甚至在多路复用器仅包含少量函数时依然成立。我们证明中的一个关键要素涉及一个组合问题：构造一棵具有多个叶子的树，每个叶子包含一组互不重叠的函数。对于每个叶子，存在一组输入，使得该叶子中的每个函数在这些输入上取相同值，即所有函数都被限制。我们展示了如何选择一组好的输入来有效限制这些函数，从而迫使每个叶子中的函数数量尽可能小，同时保持所有叶子中函数的总数量不变。这导致了叶子数量的大幅增加。