236,395 research outputs found
The Densest k-Subhypergraph Problem
The Densest -Subgraph (DS) problem, and its corresponding minimization
problem Smallest -Edge Subgraph (SES), have come to play a central role
in approximation algorithms. This is due both to their practical importance,
and their usefulness as a tool for solving and establishing approximation
bounds for other problems. These two problems are not well understood, and it
is widely believed that they do not an admit a subpolynomial approximation
ratio (although the best known hardness results do not rule this out).
In this paper we generalize both DS and SES from graphs to hypergraphs.
We consider the Densest -Subhypergraph problem (given a hypergraph ,
find a subset of vertices so as to maximize the number of
hyperedges contained in ) and define the Minimum -Union problem (given a
hypergraph, choose of the hyperedges so as to minimize the number of
vertices in their union). We focus in particular on the case where all
hyperedges have size 3, as this is the simplest non-graph setting. For this
case we provide an -approximation (for arbitrary constant )
for Densest -Subhypergraph and an -approximation for
Minimum -Union. We also give an -approximation for Minimum
-Union in general hypergraphs. Finally, we examine the interesting special
case of interval hypergraphs (instances where the vertices are a subset of the
natural numbers and the hyperedges are intervals of the line) and prove that
both problems admit an exact polynomial time solution on these instances.Comment: 21 page
Approximate Set Union Via Approximate Randomization
We develop an randomized approximation algorithm for the size of set union
problem \arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert, which given a list
of sets with approximate set size for with , and biased random generators
with Prob(x=\randomElm(A_i))\in \left[{1-\alpha_L\over |A_i|},{1+\alpha_R\over
|A_i|}\right] for each input set and element where . The approximation ratio for \arrowvert A_1\cup A_2\cup...\cup
A_m\arrowvert is in the range for any , where
. The complexity of the algorithm
is measured by both time complexity, and round complexity. The algorithm is
allowed to make multiple membership queries and get random elements from the
input sets in one round. Our algorithm makes adaptive accesses to input sets
with multiple rounds. Our algorithm gives an approximation scheme with
O(\setCount\cdot(\log \setCount)^{O(1)}) running time and rounds,
where is the number of sets. Our algorithm can handle input sets that can
generate random elements with bias, and its approximation ratio depends on the
bias. Our algorithm gives a flexible tradeoff with time complexity
O\left(\setCount^{1+\xi}\right) and round complexity for any
An extension of disjunctive programming and its impact for compact tree formulations
In the 1970's, Balas introduced the concept of disjunctive programming, which
is optimization over unions of polyhedra. One main result of his theory is
that, given linear descriptions for each of the polyhedra to be taken in the
union, one can easily derive an extended formulation of the convex hull of the
union of these polyhedra. In this paper, we give a generalization of this
result by extending the polyhedral structure of the variables coupling the
polyhedra taken in the union. Using this generalized concept, we derive
polynomial size linear programming formulations (compact formulations) for a
well-known spanning tree approximation of Steiner trees, for Gomory-Hu trees,
and, as a consequence, of the minimum -cut problem (but not for the
associated -cut polyhedron). Recently, Kaibel and Loos (2010) introduced a
more involved framework called {\em polyhedral branching systems} to derive
extended formulations. The most parts of our model can be expressed in terms of
their framework. The value of our model can be seen in the fact that it
completes their framework by an interesting algorithmic aspect.Comment: 17 page
An extension of disjunctive programming and its impact for compact tree formulations
In the 1970’s, Balas [2, 4] introduced the concept of disjunctive programming, which is optimization over unions of polyhedra. One main result of his theory is that, given linear descriptions for each of the polyhedra to be taken in the union, one can easily derive an extended formulation of the convex hull of the union of these polyhedra. In this paper, we give a generalization of this result by extending the polyhedral structure of the variables coupling the polyhedra taken in the union. Using this generalized concept, we derive polynomial size linear programming formulations (compact formulations) of a well- known spanning tree approximation of Steiner trees and flow equivalent trees for node- as well as edge- capacitated undirected networks. We also present a compact formulation for Gomory-Hu trees, and, as a consequence, of the minimum T-cut problem (but not for the associated T-cut polyhedron). Recently, Kaibel and Loos [10] introduced a more involved framework called polyhedral branching systems to derive extended formulations. The most of our model can be expressed in terms of their framework. The value of our model can be seen in the fact that it completes their framework with an interesting algorithmic aspect.disjunctive programming, compact formulation, flow-equivalent trees, Gomory-Hu trees
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