The primal-dual scheme has been used to provide approximation algorithms for many problems. Goemans and Williamson gave a (2 − 1/(n − 1))-approximation for the Prize-Collecting Steiner Tree Problem that runs in O(n 3 log n) time-it applies the primaldual scheme once for each of the n vertices of the graph. We present a primal-dual algorithm that runs in O(n 2 log n), as it applies this scheme only once, and achieves the slightly better ratio of (2 − 2/n). We also show a tight example for the analysis of the algorithm and discuss briefly a couple of other algorithms described in the literature.

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The terminal Steiner tree problem (TST) consists of finding a minimum cost Steiner tree where each terminal is a leaf. We describe a factor 2ρ − ρ/(3ρ − 2) approximation algorithm for the TST, where ρ is the approximation factor of a given algorithm for the Steiner tree problem. Considering the current best value of ρ, this improves a previous 3.10 factor to 2.52. For the TST restricted to instances where all edge costs are either 1 or 2, we improve the approximation factor from 1.60 to 1.42.

The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to graphs whose vertices have degree one or three. In this paper, we prove the scissors congruence conjecture, posed by Haase and McAllister, for this class of polytopes.The key ingredient in the proofs is the nearest neighbor interchange on graphs and a naturally arising piecewise unimodular transformation. * FAPESP 13/03447-6 and 15/10323-7, CNPq 456792/2014-7 and 452507/2016-2, CAPES PROEX.