Improve lower bounds for 2- and 3-local Euclidean maximum matchings

Determine stronger lower bounds for mu_2 and mu_3, where mu_k is the infimum over all finite planar point sets of the ratio between the total length of any k-local maximum Euclidean perfect matching and the total length of a global maximum Euclidean perfect matching, improving upon the current bounds mu_2 >= sqrt(3/7) and mu_3 >= sqrt(3)/2.

Background

The paper studies how well k-local maximum Euclidean perfect matchings approximate global maximum matchings on point sets in the plane. It defines mu_k as the worst-case ratio (over all point sets) of the length of a k-local maximum matching to that of a global maximum matching.

A general, metric-independent bound gives mu_k >= (k-1)/k. Using geometric arguments, the authors improve these for the plane: mu_2 >= sqrt(3/7) and mu_3 >= sqrt(3)/2, with corresponding examples showing mu_2 < 0.93 and mu_3 < 0.98. They further express the belief that the true ratio for k=3 is closer to 0.98 than to sqrt(3)/2, motivating sharper lower bounds.

The explicit open problem invites further geometric techniques to tighten these lower bounds for k=2 and k=3 beyond the established results.