Fitting Distances by Tree Metrics with Increment Error

The L p-min increment t and L p-min increment ultrametric t problems are two popular optimization problems arising from distance methods for reconstructing phyloge-netic trees. This paper gives the following results: 1. An O(n 2) approximation algorithm with ratio-3 for L 1-min increment t. 2. A ratio-O(n 1=p) approximation algorithm for L p-min increment ultrametric t. 3. The neighbor-joining algorithm can correctly reconstruct a phylogenetic tree T when increment errors are small enough under L 1-norm.