同时几何嵌入（Simultaneous Geometric Embedding，SGE）问题关注：给定一组共享同一顶点集V的图，是否存在V在平面上的一种嵌入方式，使得每幅给定的图都能以直线边绘制且无交叉。已知SGE是∃ℝ完全问题，即该问题与判定带整数系数的多项式方程组和不等式组是否存在实数解的问题在多项式时间下等价。我们证明，当输入图为边不相交（即图集合中不存在所谓的公共边）时，SGE仍然是∃ℝ完全的。\n作为中间结果，我们证明判定一条无重复边的有向游走是否可实现是∃ℝ完全的。其中，有向游走由一个顶点可重复的序列（即游走）和一个函数组成，该函数规定游走在每个内部位置应当左转还是右转。若存在顶点在平面上的嵌入，使得嵌入后的游走向导符合给定的方向，则称该有向游走是可实现的。此前已知，对于每条边最多重复336次的有向游走，其可实现性判定是∃ℝ完全的。\n这一结果解答了Schaefer在《On the Complexity of Some Geometric Problems With Fixed Parameters》（JGAA 2021）中提出的两个问题。
英文摘要：
Simultaneous Geometric Embedding (SGE) asks whether, for a given collection of graphs on the same vertex set V, there is an embedding of V in the plane that admits a crossing-free drawing with straightline edges for each of the given graphs. It is known that SGE is $\exists\mathbb{R}$-complete, that is, the problem is polynomially equivalent to deciding whether a system of polynomial equations and inequalities with integer coefficients has a real solution. We prove that SGE remains $\exists\mathbb{R}$-complete for edge-disjoint input graphs, that is, for collections of graphs without so-called public edges.
As an intermediate result, we prove that it is $\exists\mathbb{R}$-complete to decide whether a directional walk without repeating edges is realizable. Here, a directional walk consists of a sequence of not-necessarily distinct vertices (a walk) and a function prescribing for each inner position whether the walk shall turn left or shall turn right. A directional walk is realizable, if there is an embedding of its vertices in the plane such that the embedded walk turns according to the given directions. Previously it was known that realization is $\exists\mathbb{R}$-complete to decide for directional walks repeating each edge at most 336 times.
This answers two questions posed by Schaefer ["On the Complexity of Some Geometric Problems With Fixed Parameters", JGAA 2021].
Concept2Box: Joint Geometric Embeddings for Learning Two-View Knowledge Graphs
University of California, Los Angeles（加州大学洛杉矶分校）；Amazon.com Inc（亚马逊公司）；University of California, San Diego（加州大学圣地亚哥分校）；Carnegie Mellon University（卡内基梅隆大学）
