Circulant TSP: Vertices of the Edge-Length Polytope and Superpolynomial Lower Bounds

We study the edge-length polytope, motivated both by algorithmic research on the Circulant Traveling Salesman Problem (Circulant TSP) and number-theoretic research related to the Buratti-Horak-Rosa conjecture. Circulant TSP is a special case of TSP whose overall complexity is a significant still-open question, and where on an input with vertices $\{1, 2, ..., n\}$, the cost of an edge $\{i, j\}$ depends only on its length $\min\{|i-j|, n-|i-j|\}$. The edge-length polytope provides one path to solving circulant TSP instances, and we show that it is intimately connected to the factorization of $n$: the number of vertices scales with $n$ whenever $n$ is prime and with $n^{3/2}$ whenever $n$ is a prime-squared, but there are a superpolynomial number of vertices whenever $n$ is a power of 2. In contrast, the more-standard Symmetric TSP Polytope has roughly $n!$ vertices. Hence, for Circulant TSP, a brute-force algorithm checking every vertex is actually efficient in some cases, based on the factorization of $n$. As an intermediate step, we give superpolynomial lower-bounds on two combinatorial sequences related to the Buratti-Horak-Rosa conjecture, which asks what combinations of edge lengths can comprise a Hamiltonian path.