5 research outputs found
Anonymity-Preserving Space Partitions
We consider a multidimensional space partitioning problem, which we call Anonymity-Preserving Partition. Given a set P of n points in ?^d and a collection H of m axis-parallel hyperplanes, the hyperplanes of H partition the space into an arrangement A(H) of rectangular cells. Given an integer parameter t > 0, we call a cell C in this arrangement deficient if 0 < |C ? P| < t; that is, the cell contains at least one but fewer than t data points of P. Our problem is to remove the minimum number of hyperplanes from H so that there are no deficient cells. We show that the problem is NP-complete for all dimensions d ? 2. We present a polynomial-time d-approximation algorithm, for any fixed d, and we also show that the problem can be solved exactly in time (2d-0.924)^k m^O(1) + O(n), where k is the solution size. The one-dimensional case of the problem, where all hyperplanes are parallel, can be solved optimally in polynomial time, but we show that a related Interval Anonymity problem is NP-complete even in one dimension
Parameterized Complexity of Fair Bisection: FPT-Approximation meets Unbreakability
In the Minimum Bisection problem, input is a graph and the goal is to
partition the vertex set into two parts and , such that and the number of edges between and is minimized. This problem
can be viewed as a clustering problem where edges represent similarity, and the
task is to partition the vertices into two equally sized clusters, while
minimizing the number of pairs of similar objects that end up in different
clusters. In this paper, we initiate the study of a fair version of Minimum
Bisection. In this problem, the vertices of the graph are colored using one of
colors. The goal is to find a bisection with at most
edges between the parts, such that for each color , has exactly
vertices of color .
We first show that Fair Bisection is [1]-hard parameterized by even
when . On the other hand, our main technical contribution shows that is
that this hardness result is simply a consequence of the very strict
requirement that each color class has {\em exactly} vertices in .
In particular, we give an time algorithm that finds a
balanced partition with at most edges between them, such that for
each color , there are at most vertices of color
in . Our approximation algorithm is best viewed as a proof of concept
that the technique introduced by [Lampis, ICALP '18] for obtaining
FPT-approximation algorithms for problems of bounded tree-width or clique-width
can be efficiently exploited even on graphs of unbounded width. The key insight
is that the technique of Lampis is applicable on tree decompositions with
unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing
'14]). Along the way, we also derive a combinatorial result regarding tree
decompositions of graphs.Comment: Full version of ESA 2023 paper. Abstract shortened to meet the
character limi