27,826 research outputs found
Covering Points by Disjoint Boxes with Outliers
For a set of n points in the plane, we consider the axis--aligned (p,k)-Box
Covering problem: Find p axis-aligned, pairwise-disjoint boxes that together
contain n-k points. In this paper, we consider the boxes to be either squares
or rectangles, and we want to minimize the area of the largest box. For general
p we show that the problem is NP-hard for both squares and rectangles. For a
small, fixed number p, we give algorithms that find the solution in the
following running times:
For squares we have O(n+k log k) time for p=1, and O(n log n+k^p log^p k time
for p = 2,3. For rectangles we get O(n + k^3) for p = 1 and O(n log n+k^{2+p}
log^{p-1} k) time for p = 2,3.
In all cases, our algorithms use O(n) space.Comment: updated version: - changed problem from 'cover exactly n-k points' to
'cover at least n-k points' to avoid having non-feasible solutions. Results
are unchanged. - added Proof to Lemma 11, clarified some sections - corrected
typos and small errors - updated affiliations of two author
On three soft rectangle packing problems with guillotine constraints
We investigate how to partition a rectangular region of length and
height into rectangles of given areas using
two-stage guillotine cuts, so as to minimize either (i) the sum of the
perimeters, (ii) the largest perimeter, or (iii) the maximum aspect ratio of
the rectangles. These problems play an important role in the ongoing Vietnamese
land-allocation reform, as well as in the optimization of matrix multiplication
algorithms. We show that the first problem can be solved to optimality in
, while the two others are NP-hard. We propose mixed
integer programming (MIP) formulations and a binary search-based approach for
solving the NP-hard problems. Experimental analyses are conducted to compare
the solution approaches in terms of computational efficiency and solution
quality, for different objectives
Revisiting Numerical Pattern Mining with Formal Concept Analysis
In this paper, we investigate the problem of mining numerical data in the
framework of Formal Concept Analysis. The usual way is to use a scaling
procedure --transforming numerical attributes into binary ones-- leading either
to a loss of information or of efficiency, in particular w.r.t. the volume of
extracted patterns. By contrast, we propose to directly work on numerical data
in a more precise and efficient way, and we prove it. For that, the notions of
closed patterns, generators and equivalent classes are revisited in the
numerical context. Moreover, two original algorithms are proposed and used in
an evaluation involving real-world data, showing the predominance of the
present approach
Tight Hardness Results for Maximum Weight Rectangles
Given weighted points (positive or negative) in dimensions, what is
the axis-aligned box which maximizes the total weight of the points it
contains?
The best known algorithm for this problem is based on a reduction to a
related problem, the Weighted Depth problem [T. M. Chan, FOCS'13], and runs in
time . It was conjectured [Barbay et al., CCCG'13] that this runtime is
tight up to subpolynomial factors. We answer this conjecture affirmatively by
providing a matching conditional lower bound. We also provide conditional lower
bounds for the special case when points are arranged in a grid (a well studied
problem known as Maximum Subarray problem) as well as for other related
problems.
All our lower bounds are based on assumptions that the best known algorithms
for the All-Pairs Shortest Paths problem (APSP) and for the Max-Weight k-Clique
problem in edge-weighted graphs are essentially optimal
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