This is a tribute to my dear life-long friend, mentor and colleague Ted Swart. It includes anecdotal stories and memories of our times together, and also includes a new academic contribution in his honour, Teds polytope. Tweeks made to the Birkhoff polytope Bn endow Teds polytope Tn({\epsilon}) with a special tunable parameter {\epsilon} = {\epsilon}(n). Observe how Bn can be viewed as the convex hull of both the TSP polytope, and the set of non-tour permutation extrema, and, that its extended formulation is compact. Tours (connected 2-factor permutation matrices when viewed as adjacency matrices) can be distinguished from non-tours (disconnected 2-factor permutation matrices) where {\epsilon} scales the magnitude of tweeks made to Bn. For {\epsilon} > 0, Tn({\epsilon}) is tuned so that the convex hull of extrema corresponding to transformed tours is lifted from Bn, and separated (by a hyperplane) from the convex hull of extrema corresponding to translated non-tours. This leads to creation of the feasible region of an LP model that can decide existence of a tour in a graph based on an extended formulation of the TSP polytope. That is, by designing for polynomial-time distinguishable tour extrema embedded in a subspace disjoint from non-tour extrema, NP-completeness strongholds come into play, necessarily expressed in a non-compact extended formulation of Tn({\epsilon}) i.e. a compact extended formulation of the TSP polytope cannot exist. No matter, Ted would have loved these ideas, and Tn({\epsilon}) might one day yet be useful in the study of the P versus NP conundrum. In summary, Tn({\epsilon}) is a perturbed Bn i.e. the convex hull of both an {\epsilon}-stretched TSP polytope, and the set of translated non-tour permutation extrema i.e. a TSP-like polytope and separable non-tour extrema.

翻译：本文是对我挚爱的终身挚友、导师兼同事泰德·斯沃特的致敬。文中收录了我们共同度过的轶事与回忆，并包含一项以他命名的全新学术贡献——泰德多面体。通过对伯克霍夫多面体Bn进行微调，赋予泰德多面体Tn(ε)一个特殊的可调参数ε = ε(n)。观察发现，Bn可被视为TSP多面体与非旅行排列极值点集合的凸包，且其扩展形式是紧的。旅行（作为邻接矩阵时对应连通2-因子置换矩阵）可与非旅行（非连通2-因子置换矩阵）相区分，其中ε用于缩放对Bn施加微调的程度。当ε > 0时，Tn(ε)经过调谐，使得对应变换后旅行排列的极值点凸包从Bn中提升，并通过超平面与对应平移后非旅行排列的极值点凸包分离。这构造了一个线性规划模型的可行域，该模型能基于TSP多面体的扩展形式判定图中是否存在哈马尔顿回路。换言之，通过设计嵌入子空间的可多项式时间区分的旅行极值点（与非旅行极值点不相交），NP完全性难题必然显现，并以Tn(ε)的非紧扩展形式表达——即TSP多面体不存在紧扩展形式。但无论如何，泰德定会喜爱这些构想，且Tn(ε)或许终有一日能用于P对NP难题的研究。简言之，Tn(ε)是扰动后的Bn，即ε-拉伸的TSP多面体与平移后非旅行置换极值点集合的凸包——一个类TSP多面体与可分离的非旅行极值点。