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On two segmentation problems Benny Sudakov

Summary: On two segmentation problems
Noga Alon
Benny Sudakov
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
i=1 cSi
Pi 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) 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


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics