321 research outputs found
Faster SDP hierarchy solvers for local rounding algorithms
Convex relaxations based on different hierarchies of linear/semi-definite
programs have been used recently to devise approximation algorithms for various
optimization problems. The approximation guarantee of these algorithms improves
with the number of {\em rounds} in the hierarchy, though the complexity of
solving (or even writing down the solution for) the 'th level program grows
as where is the input size.
In this work, we observe that many of these algorithms are based on {\em
local} rounding procedures that only use a small part of the SDP solution (of
size instead of ). We give an algorithm to
find the requisite portion in time polynomial in its size. The challenge in
achieving this is that the required portion of the solution is not fixed a
priori but depends on other parts of the solution, sometimes in a complicated
iterative manner.
Our solver leads to time algorithms to obtain the same
guarantees in many cases as the earlier time algorithms based on
rounds of the Lasserre hierarchy. In particular, guarantees based on rounds can be realized in polynomial time.
We develop and describe our algorithm in a fairly general abstract framework.
The main technical tool in our work, which might be of independent interest in
convex optimization, is an efficient ellipsoid algorithm based separation
oracle for convex programs that can output a {\em certificate of infeasibility
with restricted support}. This is used in a recursive manner to find a sequence
of consistent points in nested convex bodies that "fools" local rounding
algorithms.Comment: 30 pages, 8 figure
Engineering Branch-and-Cut Algorithms for the Equicut Problem
A minimum equicut of an edge-weighted graph is a partition of the nodes of the graph into two sets of equal size such hat the sum of the weights of edges joining nodes in different partitions is minimum. We compare basic linear and semidefnite relaxations for the equicut problem, and and that linear bounds are competitive with the corresponding semidefnite ones but can be computed much faster. Motivated by an application of equicut in theoretical physics, we revisit an approach by Brunetta et al. and present an enhanced branch-and-cut algorithm. Our computational results suggest that the proposed branch-andcut algorithm has a better performance than the algorithm of Brunetta et al.. Further, it is able to solve to optimality in reasonable time several instances with more than 200 nodes from the physics application
Polytopal complexes: maps, chain complexes and... necklaces
The notion of polytopal map between two polytopal complexes is defined.
Surprisingly, this definition is quite simple and extends naturally those of
simplicial and cubical maps. It is then possible to define an induced chain map
between the associated chain complexes. Finally, we use this new tool to give
the first combinatorial proof of the splitting necklace theorem of Alon. The
paper ends with open questions, such as the existence of Sperner's lemma for a
polytopal complex or the existence of a cubical approximation theorem.Comment: Presented at the TGGT 08 Conference, May 2008, Paris. The definition
of a polytopal map has been modifie
Linear Programming and Community Detection
The problem of community detection with two equal-sized communities is
closely related to the minimum graph bisection problem over certain random
graph models. In the stochastic block model distribution over networks with
community structure, a well-known semidefinite programming (SDP) relaxation of
the minimum bisection problem recovers the underlying communities whenever
possible. Motivated by their superior scalability, we study the theoretical
performance of linear programming (LP) relaxations of the minimum bisection
problem for the same random models. We show that unlike the SDP relaxation that
undergoes a phase transition in the logarithmic average-degree regime, the LP
relaxation exhibits a transition from recovery to non-recovery in the linear
average-degree regime. We show that in the logarithmic average-degree regime,
the LP relaxation fails in recovering the planted bisection with high
probability.Comment: 35 pages, 3 figure
Community Detection: Statistical Inference Models
Community detection in large networks through the methods based on the statistical inference model can identify the node community as well as find the interaction between the communities. Statistical inference based methods try to fit a generative model to the network data. This paper discusses the statistical inference methods which groups the communities on vertices or nodes
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