On Two Segmentation Problems

The hypercube segmentation problem is the following: Given a set S of m vertices of the discrete d-dimensional cube {0,1}d, find k vertices P1,?,Pk, Pi?{0,1}d and a partitions of S into k segments S1,?,Sk so as to maximize the sum ?ki=1?c?SiPi?c, where ? is the overlap operator between two vertices of the d-dimensional cube, defined to be the number of positions they have in common. This problem (among other ones) was considered by Kleinberg, Papadimitriou, and Raghavan, where the authors designed a deterministic approximation algorithm that runs in polynomial time for every fixed k and produces a solution whose value is within 0.828 of the optimum, as well as a randomized algorithm that runs in linear time for every fixed k and produces a solution whose expected value is within 0.7 of the optimum. Here we design an improved approximation algorithm; for every fixed ?0 and every fixed k our algorithm produces in linear time a solution whose value is within (1??) of the optimum. Therefore, for every fixed k, this is a polynomial approximation scheme for the problem. The algorithm is deterministic, but it is convenient to first describe it as a randomized algorithm and then to derandomize it using some properties of expander graphs. We also consider a segmented version of the minimum spanning tree problem, where we show that no approximation can be achieved in polynomial time, unless P=NP.