Algorithms for a Simple Point Placement Problem

We consider algorithms for a simple one-dimensionalpoint placement problem: given N points on a line, and noisy measurements of the distances between many pairs of them, estimate the relative positions of the points. Problems of this flavor arise in a variety of contexts. The particular motivating example that inspired this work comes from molecular biology; the points are markers on a chromosome and the goal is to map their positions. The problem is NP-hard under reasonable assumptions. We present two algorithms for computing least squares estimates of the ordering and positions of the markers: a branch and bound algorithm and a highly effective heuristic search algorithm. The branch and bound algorithm is able to solve to optimality problems of 18 markers in about an hour, visiting about 106 nodes out of a search space of 1016 nodes. The local search algorithm usually was able to find the global minimum of problems of similar size in about one second, and should comfortably handle much larger problem instances.