89,603 research outputs found
A QPTAS for Maximum Weight Independent Set of Polygons with Polylogarithmically Many Vertices
The Maximum Weight Independent Set of Polygons problem is a fundamental
problem in computational geometry. Given a set of weighted polygons in the
2-dimensional plane, the goal is to find a set of pairwise non-overlapping
polygons with maximum total weight. Due to its wide range of applications, the
MWISP problem and its special cases have been extensively studied both in the
approximation algorithms and the computational geometry community. Despite a
lot of research, its general case is not well-understood. Currently the best
known polynomial time algorithm achieves an approximation ratio of n^(epsilon)
[Fox and Pach, SODA 2011], and it is not even clear whether the problem is
APX-hard. We present a (1+epsilon)-approximation algorithm, assuming that each
polygon in the input has at most a polylogarithmic number of vertices. Our
algorithm has quasi-polynomial running time.
We use a recently introduced framework for approximating maximum weight
independent set in geometric intersection graphs. The framework has been used
to construct a QPTAS in the much simpler case of axis-parallel rectangles. We
extend it in two ways, to adapt it to our much more general setting. First, we
show that its technical core can be reduced to the case when all input polygons
are triangles. Secondly, we replace its key technical ingredient which is a
method to partition the plane using only few edges such that the objects
stemming from the optimal solution are evenly distributed among the resulting
faces and each object is intersected only a few times. Our new procedure for
this task is not more complex than the original one, and it can handle the
arising difficulties due to the arbitrary angles of the polygons. Note that
already this obstacle makes the known analysis for the above framework fail.
Also, in general it is not well understood how to handle this difficulty by
efficient approximation algorithms
The Communication Complexity of Set Intersection and Multiple Equality Testing
In this paper we explore fundamental problems in randomized communication
complexity such as computing Set Intersection on sets of size and Equality
Testing between vectors of length . Sa\u{g}lam and Tardos and Brody et al.
showed that for these types of problems, one can achieve optimal communication
volume of bits, with a randomized protocol that takes
rounds. Aside from rounds and communication volume, there is a \emph{third}
parameter of interest, namely the \emph{error probability} .
It is straightforward to show that protocols for Set Intersection or Equality
Testing need to send bits. Is it
possible to simultaneously achieve optimality in all three parameters, namely
communication and rounds? In
this paper we prove that there is no universally optimal algorithm, and
complement the existing round-communication tradeoffs with a new tradeoff
between rounds, communication, and probability of error. In particular:
1. Any protocol for solving Multiple Equality Testing in rounds with
failure probability has communication volume .
2. There exists a protocol for solving Multiple Equality Testing in rounds with communication, thereby essentially
matching our lower bound and that of Sa\u{g}lam and Tardos.
Our original motivation for considering as an independent
parameter came from the problem of enumerating triangles in distributed
() networks having maximum degree . We prove that
this problem can be solved in time with
high probability .Comment: 44 page
A Finite-Time Cutting Plane Algorithm for Distributed Mixed Integer Linear Programming
Many problems of interest for cyber-physical network systems can be
formulated as Mixed Integer Linear Programs in which the constraints are
distributed among the agents. In this paper we propose a distributed algorithm
to solve this class of optimization problems in a peer-to-peer network with no
coordinator and with limited computation and communication capabilities. In the
proposed algorithm, at each communication round, agents solve locally a small
LP, generate suitable cutting planes, namely intersection cuts and cost-based
cuts, and communicate a fixed number of active constraints, i.e., a candidate
optimal basis. We prove that, if the cost is integer, the algorithm converges
to the lexicographically minimal optimal solution in a finite number of
communication rounds. Finally, through numerical computations, we analyze the
algorithm convergence as a function of the network size.Comment: 6 pages, 3 figure
Innovation Pursuit: A New Approach to Subspace Clustering
In subspace clustering, a group of data points belonging to a union of
subspaces are assigned membership to their respective subspaces. This paper
presents a new approach dubbed Innovation Pursuit (iPursuit) to the problem of
subspace clustering using a new geometrical idea whereby subspaces are
identified based on their relative novelties. We present two frameworks in
which the idea of innovation pursuit is used to distinguish the subspaces.
Underlying the first framework is an iterative method that finds the subspaces
consecutively by solving a series of simple linear optimization problems, each
searching for a direction of innovation in the span of the data potentially
orthogonal to all subspaces except for the one to be identified in one step of
the algorithm. A detailed mathematical analysis is provided establishing
sufficient conditions for iPursuit to correctly cluster the data. The proposed
approach can provably yield exact clustering even when the subspaces have
significant intersections. It is shown that the complexity of the iterative
approach scales only linearly in the number of data points and subspaces, and
quadratically in the dimension of the subspaces. The second framework
integrates iPursuit with spectral clustering to yield a new variant of
spectral-clustering-based algorithms. The numerical simulations with both real
and synthetic data demonstrate that iPursuit can often outperform the
state-of-the-art subspace clustering algorithms, more so for subspaces with
significant intersections, and that it significantly improves the
state-of-the-art result for subspace-segmentation-based face clustering
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