We initiate the focused study of constant-cost randomized communication, with emphasis on its connection to graph representations. We observe that constant-cost randomized communication problems are equivalent to hereditary (i.e. closed under taking induced subgraphs) graph classes which admit constant-size adjacency sketches and probabilistic universal graphs (PUGs), which are randomized versions of the well-studied adjacency labeling schemes and induced-universal graphs. This gives a new perspective on long-standing questions about the existence of these objects, including new methods of constructing adjacency labeling schemes. We ask three main questions about constant-cost communication, or equivalently, constant-size PUGs: (1) Are there any natural, non-trivial problems aside from Equality and k-Hamming Distance which have constant-cost communication? We provide a number of new examples, including deciding whether two vertices have path-distance at most k in a planar graph, and showing that constant-size PUGs are preserved by the Cartesian product operation. (2) What structures of a problem explain the existence or non-existence of a constant-cost protocol? We show that in many cases a Greater-Than subproblem is such a structure. (3) Is the Equality problem complete for constant-cost randomized communication? We show that it is not: there are constant-cost problems which do not reduce to Equality.

翻译：我们首次系统研究了恒定代价随机化通信，重点探讨其与图表示的联系。我们观察到恒定代价随机化通信问题等价于满足以下条件的遗传性图类（即在导出子图操作下封闭）：该类图允许恒定规模的邻接草图与概率通用图（PUGs）——这两者分别是经过深入研究的邻接标记方案与导出通用图的随机化版本。这为关于这些对象存在性的长期问题提供了新视角，包括构造邻接标记方案的新方法。针对恒定代价通信（即恒定规模PUGs），我们提出三个核心问题：（1）除相等性判定与k-汉明距离外，是否存在其他自然且非平凡的恒定代价通信问题？我们给出了若干新示例，包括判定平面图中两顶点路径距离是否不超过k，并证明笛卡尔积运算能保持恒定规模PUGs的存在性。（2）问题的何种结构能解释恒定代价协议的存在与否？我们证明在多数情况下，“大于比较”子问题正是此类关键结构。（3）相等性判定问题是否为恒定代价随机化通信的完全问题？我们证明并非如此：存在不归约于相等性判定的恒定代价问题。