On the turnpike problem

The turnpike problem, also known as the partial digest problem , is: Given a multiset of ( n2 ) positive numbers ΔX, does there exist a set X such that ΔX is exactly the multiset of all positive pairwise differences of the elements of X. The complexity of the problem is not known. We write the turnpike problem as a 0–1 quadratic program. In order to solve a quadratic program, we relax it to a semidefinite program, which can be solved in polynomial time. We give three different formulations of the turnpike problem as a 0–1 quadratic program. For the first 0–1 quadratic program we introduce a sequence of semidefinite relaxations, similar to the sequence of semidefinite relaxations proposed by Lovasz and Schrijver in their seminal paper “Cones of matrices and set-functions and 0–1 optimization” ( SIAM Journal on Optimization 1, pp 166–190, 1990). Although a powerful tool, this method has not been used except in their original paper to develop a polynomial time algorithm for finding stable sets in perfect graphs. We give some theoretical results on these relaxations and show how they can be used to solve the turnpike problem in polynomial time for some classes of instances. These classes include the class of instances constructed by Zhang in his paper “An exponential example for partial digest mapping algorithm” (Tech Report, Computer Science Dept., Penn State University 1993) and the class of instances that have a unique solution and all the numbers in ΔX are different and on which Skiena, Smith and Lemke's backtracking procedure, from their paper “Reconstructing sets from interpoint distances” (Proc. Sixth ACM Symp. Computational Geometry, pp 332–339, 1990) backtracks only a constant number of steps. Previously it was not known how to solve the former in polynomial time. We use our theoretical formulations to develop a polynomial time heuristic to solve general instances of the problem. We perform extensive numerical testing of our methods. To date we do not have an instance of the turnpike problem for which our methods do not yield a solution. The second 0–1 quadratic program formulation of the turnpike problem will be too large for practical purposes. We use association schemes and some other methods to reduce its size and obtain the third 0–1 quadratic program. We establish a connection between this relaxation and the first relaxation and show its limitations.