On Two Geometric Problems Related to the Traveling Salesman Problem

Abstract The degree- K Minimum Spanning Tree (MST) problem asks for the minimum length spanning tree that has no vertex of degree greater than K . The Euclidean degree- K MST problem is known to be tractable for K ⩾ 5; the degree-2 MST is simply the Euclidean path-TSP, which is NP-complete. It is proved that the Euclidean degree-3 MST problem is also NP-complete, thus leaving open only the case for K = 4. Among the most illustrious approximation algorithms is the heuristic for the Euclidean TSP due to Christofides. It is proved that implementing the “shortcutting phase” of Christofides' algorithm optimally is NP-hard (even so, Christofides' algorithm guarantees a tour which is no more than 50% longer than the optimal one).