Is there a local variant of TSP? I'm a traveling salesman and I have n days to sell, I can start anywhere, I can sell once per city. I want to know where to start and what route to take.

It's likely NP-hard, I was just wondering if there's a name for this or similar variants on TSP.

EDIT

Here's a clarification of the problem:


Let $S$ be a clique with weighted edges that one could put onto a plane (Euclidian distances, etc.) and $k$ be a positive real number.
A solution $G$ is a subgraph of $S$ that is a simple path such that $\Sigma_{e \in E(G)} e &lt; k$ and $|V(G)|$ is maximized.


Or, in terms of geometry:


Let $S$ be a set of points on the plane
A solution is a path through some of the points of length less than $k$ that maximizes the number of points hit.
This is an "Orienteering problem" and is NP-hard [1]. If the starting vertex is given the problem is rooted; your variant is unrooted. Some authors call it the "selective travelling salesman" [2].

[1] B. L. Golden, L. Levy, and R. Vohra. "The orienteering problem". In: Naval Research Logistics 34.3 (1987), pp. 307-­318.

[2] M. Gendreau, G. Laporte, and F. Semet. "A tabu search heuristic for the undirected selective travelling salesman problem". In: European Journal of Operational Research (1998), pp. 539­-545.