4,968 research outputs found
Almost Symmetries and the Unit Commitment Problem
This thesis explores two main topics. The first is almost symmetry detection on graphs. The presence of symmetry in combinatorial optimization problems has long been considered an anathema, but in the past decade considerable progress has been made. Modern integer and constraint programming solvers have automatic symmetry detection built-in to either exploit or avoid symmetric regions of the search space. Automatic symmetry detection generally works by converting the input problem to a graph which is in exact correspondence with the problem formulation. Symmetry can then be detected on this graph using one of the excellent existing algorithms; these are also the symmetries of the problem formulation.The motivation for detecting almost symmetries on graphs is that almost symmetries in an integer program can force the solver to explore nearly symmetric regions of the search space. Because of the known correspondence between integer programming formulations and graphs, this is a first step toward detecting almost symmetries in integer programming formulations. Though we are only able to compute almost symmetries for graphs of modest size, the results indicate that almost symmetry is definitely present in some real-world combinatorial structures, and likely warrants further investigation.The second topic explored in this thesis is integer programming formulations for the unit commitment problem. The unit commitment problem involves scheduling power generators to meet anticipated energy demand while minimizing total system operation cost. Today, practitioners usually formulate and solve unit commitment as a large-scale mixed integer linear program.The original intent of this project was to bring the analysis of almost symmetries to the unit commitment problem. Two power generators are almost symmetric in the unit commitment problem if they have almost identical parameters. Along the way, however, new formulations for power generators were discovered that warranted a thorough investigation of their own. Chapters 4 and 5 are a result of this research.Thus this work makes three contributions to the unit commitment problem: a convex hull description for a power generator accommodating many types of constraints, an improved formulation for time-dependent start-up costs, and an exact symmetry reduction technique via reformulation
Breaking Instance-Independent Symmetries In Exact Graph Coloring
Code optimization and high level synthesis can be posed as constraint
satisfaction and optimization problems, such as graph coloring used in register
allocation. Graph coloring is also used to model more traditional CSPs relevant
to AI, such as planning, time-tabling and scheduling. Provably optimal
solutions may be desirable for commercial and defense applications.
Additionally, for applications such as register allocation and code
optimization, naturally-occurring instances of graph coloring are often small
and can be solved optimally. A recent wave of improvements in algorithms for
Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests
generic problem-reduction methods, rather than problem-specific heuristics,
because (1) heuristics may be upset by new constraints, (2) heuristics tend to
ignore structure, and (3) many relevant problems are provably inapproximable.
Problem reductions often lead to highly symmetric SAT instances, and
symmetries are known to slow down SAT solvers. In this work, we compare several
avenues for symmetry breaking, in particular when certain kinds of symmetry are
present in all generated instances. Our focus on reducing CSPs to SAT allows us
to leverage recent dramatic improvement in SAT solvers and automatically
benefit from future progress. We can use a variety of black-box SAT solvers
without modifying their source code because our symmetry-breaking techniques
are static, i.e., we detect symmetries and add symmetry breaking predicates
(SBPs) during pre-processing.
An important result of our work is that among the types of
instance-independent SBPs we studied and their combinations, the simplest and
least complete constructions are the most effective. Our experiments also
clearly indicate that instance-independent symmetries should mostly be
processed together with instance-specific symmetries rather than at the
specification level, contrary to what has been suggested in the literature
Detecting degree symmetries in networks
The surrounding of a vertex in a network can be more or less symmetric. We
derive measures of a specific kind of symmetry of a vertex which we call degree
symmetry -- the property that many paths going out from a vertex have
overlapping degree sequences. These measures are evaluated on artificial and
real networks. Specifically we consider vertices in the human metabolic
network. We also measure the average degree-symmetry coefficient for different
classes of real-world network. We find that most studied examples are weakly
positively degree-symmetric. The exceptions are an airport network (having a
negative degree-symmetry coefficient) and one-mode projections of social
affiliation networks that are rather strongly degree-symmetric
Quantum automorphisms of folded cube graphs
We show that the quantum automorphism group of the Clebsch graph is
. This answers a question by Banica, Bichon and Collins from 2007.
More general for odd , the quantum automorphism group of the folded -cube
graph is . Furthermore, we show that if the automorphism group of a
graph contains a pair of disjoint automorphisms this graph has quantum
symmetry.Comment: 19 pages, corrected a mistake in the proof of Theorem 2.2 + minor
change
Algorithms for Highly Symmetric Linear and Integer Programs
This paper deals with exploiting symmetry for solving linear and integer
programming problems. Basic properties of linear representations of finite
groups can be used to reduce symmetric linear programming to solving linear
programs of lower dimension. Combining this approach with knowledge of the
geometry of feasible integer solutions yields an algorithm for solving highly
symmetric integer linear programs which only takes time which is linear in the
number of constraints and quadratic in the dimension.Comment: 21 pages, 1 figure; some references and further comments added, title
slightly change
Tests of Higgs and Top Effective Interactions
We study the possibility to detect heavy physics effects in the interactions
of Higgs bosons and the top quark at future colliders using the effective
Lagrangian approach. The modification of the interactions may enhance the
production of Higgs bosons at hadron colliders through the mechanisms of gluon
fusion and associated production with a W boson or pairs. The most
promising signature is through the decay of the Higgs boson into two photons,
whose branching ratio is also enhanced in this approach. As a consequence of
our analysis we get a bound on the chromomagnetic dipole moment of the top
quark.Comment: 14 pages, Latex, two figures available by fax under request. To be
published in Phys. Lett
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