In Honour of Ted Swart
share
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(ϵ) with a special tunable parameter ϵ = ϵ(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 ϵ scales the magnitude of tweeks made to Bn. For ϵ > 0, Tn(ϵ) 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(ϵ) i.e. a compact extended formulation of the TSP polytope cannot exist. No matter, Ted would have loved these ideas, and Tn(ϵ) might one day yet be useful in the study of the P versus NP conundrum. In summary, Tn(ϵ) is a perturbed Bn i.e. the convex hull of both an ϵ-stretched TSP polytope, and the set of translated non-tour permutation extrema i.e. a TSP-like polytope and separable non-tour extrema.
READ FULL TEXT
