Computing Nonsimple Polygons of Minimum Perimeter

TL;DR

This paper studies the Minimum Perimeter Polygon problem, a geometric relaxation of TSP, proving its NP-hardness, providing approximation algorithms, and demonstrating practical solutions for large instances.

Contribution

It establishes NP-hardness of MPP, develops approximation methods, and introduces geometric techniques for solving large instances efficiently.

Findings

NP-hardness of MPP proven

Constant-factor approximation achieved

Exact solutions for instances with up to 600 vertices

Abstract

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.