We present a (5/3 - epsilon)-approximation algorithm for some constant
epsilon>0 for the traveling salesman path problem under the unit-weight
graphical metric, and prove an upper bound on the integrality gap of the
path-variant Held-Karp relaxation both under this metric and the general
metric. Given a complete graph with the metric cost and two designated
endpoints in the graph, the traveling salesman path problem is to find a
minimum Hamiltonian path between these two endpoints. The best previously known
performance guarantee for this problem was 5/3 and was due to Hoogeveen.