317 research outputs found
A Bundle Approach for SDPs with Exact Subgraph Constraints
The 'exact subgraph' approach was recently introduced as a hierarchical
scheme to get increasingly tight semidefinite programming relaxations of
several NP-hard graph optimization problems. Solving these relaxations is a
computational challenge because of the potentially large number of violated
subgraph constraints. We introduce a computational framework for these
relaxations designed to cope with these difficulties. We suggest a partial
Lagrangian dual, and exploit the fact that its evaluation decomposes into two
independent subproblems. This opens the way to use the bundle method from
non-smooth optimization to minimize the dual function. Computational
experiments on the Max-Cut, stable set and coloring problem show the efficiency
of this approach
On different Versions of the Exact Subgraph Hierarchy for the Stable Set Problem
Let be a graph with vertices and edges. One of several
hierarchies towards the stability number of is the exact subgraph hierarchy
(ESH). On the first level it computes the Lov\'{a}sz theta function
as semidefinite program (SDP) with a matrix variable of order
and constraints. On the -th level it adds all exact subgraph
constraints (ESC) for subgraphs of order to the SDP. An ESC ensures that
the submatrix of the matrix variable corresponding to the subgraph is in the
correct polytope. By including only some ESCs into the SDP the ESH can be
exploited computationally.
In this paper we introduce a variant of the ESH that computes
through an SDP with a matrix variable of order and constraints. We
show that it makes sense to include the ESCs into this SDP and introduce the
compressed ESH (CESH) analogously to the ESH. Computationally the CESH seems
favorable as the SDP is smaller. However, we prove that the bounds based on the
ESH are always at least as good as those of the CESH. In computations sometimes
they are significantly better.
We also introduce scaled ESCs (SESCs), which are a more natural way to
include exactness constraints into the smaller SDP and we prove that including
an SESC is equivalent to including an ESC for every subgraph
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The Internet And The American Political System
The past eight years have seen a great increase in Internet usage in American culture and politics. It would seem that, in our digital age, the Internet has exercised strong effects on political behavior and even on legislators. This thesis explores the variety and intensity of these effects, finding them to be substantial and growing, although not yet robust. The main influences the net has exerted on American politics take place predominantly within two areas: political campaigns and online political interest groups. Activists are certainly using the Internet for political causes, but this sort of Internet usage is really just an extension of previous activism. The Internet does not create new habits; it simply offers a more convenient method of reading the news, communicating to others, or performing other activities we have already been inclined to perform. Even those Internet users who access political web sites are shown preeminently to be those who have otherwise accessed political information in other ways such as newspapers or televised news. So far the Internet has made campaign donations easier for people who are comfortable surfing the World Wide Web. But there is little evidence to show that these people would not have otherwise donated to the campaign by more traditional methods. The Internet has made political activism easier, but people who are not politically active will not suddenly change simply because the Internet offers itself as an expedient, inexpensive tool. We have seen, however, with groups like MoveOn.org, that activists are rallying, communicating, and demonstrating more efficiently than ever before. The political parties or groups that can most effectively use the Internet to mobilize voters and affect public opinion will greatly benefit themselves
Sum-of-Squares Certificates for Vizing's Conjecture via Determining Gr\"obner Bases
The famous open Vizing conjecture claims that the domination number of the
Cartesian product graph of two graphs and is at least the product of
the domination numbers of and . Recently Gaar, Krenn, Margulies and
Wiegele used the graph class of all graphs with
vertices and domination number and reformulated Vizing's
conjecture as the problem that for all graph classes and
the Vizing polynomial is sum-of-squares (SOS) modulo the Vizing
ideal. By solving semidefinite programs (SDPs) and clever guessing they derived
SOS-certificates for some values of , ,
, and .
In this paper, we consider their approach for . For this case we are able to derive the unique reduced Gr\"obner basis of
the Vizing ideal. Based on this, we deduce the minimum degree of an SOS-certificate for Vizing's conjecture, which is
the first result of this kind. Furthermore, we present a method to find
certificates for graph classes and with
for general , which is again based on
solving SDPs, but does not depend on guessing and depends on much smaller SDPs.
We implement our new method in SageMath and give new SOS-certificates for all
graph classes and with
and .Comment: 36 pages, 2 figure
Relationship of -Bend and Monotonic -Bend Edge Intersection Graphs of Paths on a Grid
If a graph can be represented by means of paths on a grid, such that each
vertex of corresponds to one path on the grid and two vertices of are
adjacent if and only if the corresponding paths share a grid edge, then this
graph is called EPG and the representation is called EPG representation. A
-bend EPG representation is an EPG representation in which each path has at
most bends. The class of all graphs that have a -bend EPG representation
is denoted by . is the class of all graphs that have a
monotonic (each path is ascending in both columns and rows) -bend EPG
representation.
It is known that holds for . We prove that
holds also for and for by investigating the -membership and -membership of complete
bipartite graphs. In particular we derive necessary conditions for this
membership that have to be fulfilled by , and , where and are
the number of vertices on the two partition classes of the bipartite graph. We
conjecture that holds also for .
Furthermore we show that holds for all
. This implies that restricting the shape of the paths can lead
to a significant increase of the number of bends needed in an EPG
representation. So far no bounds on the amount of that increase were known. We
prove that holds, providing the first result of this
kind
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