The Turnpike problem: Variants, Structure, and Complexity

Abstract

We study the Turnpike problem, which is a problem of reconstructing a point set from a multiset of pairwise distances between the points. Specifically, we are interested in the combinatorial properties of the problem and its complexity. Among our combinatorial results are a method to determine the parity of the number of even length and odd length edges in the solution, a method to infer the number of odd and even coordinates, etc. In terms of complexity we discuss the place of the Turnpike problem in known complexity classes and whether Turnpike is selfreducible. We also give a lower bound on the number of queries needed to solve a newly defined version of the Turnpike problem, called OracleTurnpike. Furthermore, we investigate a variant of the Turnpike problem with some additional information. We obtained two non-trivial upper bounds on the running times of the backtracking algorithm on two restricted versions of this new variant. Finally we discuss Turnpike-like questions on trees and give NP and #P-completeness results for a related graph reconstruction problem.