  We provide exact and approximation methods for solving a geometric relaxation
of the Traveling Salesman Problem (TSP) that occurs in curve reconstruction:
for a given set of vertices in the plane, the problem Minimum Perimeter Polygon
(MPP) asks for a (not necessarily simply connected) polygon with shortest
possible boundary length. Even though the closely related problem of finding a
minimum cycle cover is polynomially solvable by matching techniques, we prove
how the topological structure of a polygon leads to NP-hardness of the MPP. On
the positive side, we show how to achieve a constant-factor approximation.
  When trying to solve MPP instances to provable optimality by means of integer
programming, an additional difficulty compared to the TSP is the fact that only
a subset of subtour constraints is valid, depending not on combinatorics, but
on geometry. We overcome this difficulty by establishing and exploiting
additional geometric properties. This allows us to reliably solve a wide range
of benchmark instances with up to 600 vertices within reasonable time on a
standard machine. We also show that using a natural geometry-based
sparsification yields results that are on average within 0.5% of the optimum.
