 
Summary: On two segmentation problems
Noga Alon
Benny Sudakov
Abstract
The hypercube segmentation problem is the following: Given a set S of m vertices of the
discrete ddimensional 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
k
i=1 cSi
Pi c,
where is the overlap operator between two vertices of the ddimensional cube, defined to be the
number of positions they have in common.
This problem (among other ones) is considered by Kleinberg, Papadimitriou and Raghavan
in [9], where the authors design 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
