9,368 research outputs found
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
Maximum weight cycle packing in directed graphs, with application to kidney exchange programs
Centralized matching programs have been established in several countries to organize kidney exchanges between incompatible patient-donor pairs. At the heart of these programs are algorithms to solve kidney exchange problems, which can be modelled as cycle packing problems in a directed graph, involving cycles of length 2, 3, or even longer. Usually, the goal is to maximize the number of transplants, but sometimes the total benefit is maximized by considering the differences between suitable kidneys. These problems correspond to computing cycle packings of maximum size or maximum weight in directed graphs. Here we prove the APX-completeness of the problem of finding a maximum size exchange involving only 2-cycles and 3-cycles. We also present an approximation algorithm and an exact algorithm for the problem of finding a maximum weight exchange involving cycles of bounded length. The exact algorithm has been used to provide optimal solutions to real kidney exchange problems arising from the National Matching Scheme for Paired Donation run by NHS Blood and Transplant, and we describe practical experience based on this collaboration
Online Circle and Sphere Packing
In this paper we consider the Online Bin Packing Problem in three variants:
Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes.
The two first ones receive an online sequence of circles (items) of different
radii while the third one receive an online sequence of spheres (items) of
different radii, and they want to pack the items into the minimum number of
unit squares, isosceles right triangles of leg length one, and unit cubes,
respectively. For Online Circle Packing in Squares, we improve the previous
best-known competitive ratio for the bounded space version, when at most a
constant number of bins can be open at any given time, from 2.439 to 2.3536.
For Online Circle Packing in Isosceles Right Triangles and Online Sphere
Packing in Cubes we show bounded space algorithms of asymptotic competitive
ratios 2.5490 and 3.5316, respectively, as well as lower bounds of 2.1193 and
2.7707 on the competitive ratio of any online bounded space algorithm for these
two problems. We also considered the online unbounded space variant of these
three problems which admits a small reorganization of the items inside the bin
after their packing, and we present algorithms of competitive ratios 2.3105,
2.5094, and 3.5146 for Circles in Squares, Circles in Isosceles Right
Triangles, and Spheres in Cubes, respectively
On largest volume simplices and sub-determinants
We show that the problem of finding the simplex of largest volume in the
convex hull of points in can be approximated with a factor
of in polynomial time. This improves upon the previously best
known approximation guarantee of by Khachiyan. On the other hand,
we show that there exists a constant such that this problem cannot be
approximated with a factor of , unless . % This improves over the
inapproximability that was previously known. Our hardness result holds
even if , in which case there exists a \bar c\,^{d}-approximation
algorithm that relies on recent sampling techniques, where is again a
constant. We show that similar results hold for the problem of finding the
largest absolute value of a subdeterminant of a matrix
Fast Convex Decomposition for Truthful Social Welfare Approximation
Approximating the optimal social welfare while preserving truthfulness is a
well studied problem in algorithmic mechanism design. Assuming that the social
welfare of a given mechanism design problem can be optimized by an integer
program whose integrality gap is at most , Lavi and Swamy~\cite{Lavi11}
propose a general approach to designing a randomized -approximation
mechanism which is truthful in expectation. Their method is based on
decomposing an optimal solution for the relaxed linear program into a convex
combination of integer solutions. Unfortunately, Lavi and Swamy's decomposition
technique relies heavily on the ellipsoid method, which is notorious for its
poor practical performance. To overcome this problem, we present an alternative
decomposition technique which yields an approximation
and only requires a quadratic number of calls to an integrality gap verifier
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