116 research outputs found
Global aspects of the space of 6D N = 1 supergravities
We perform a global analysis of the space of consistent 6D quantum gravity
theories with N = 1 supersymmetry, including models with multiple tensor
multiplets. We prove that for theories with fewer than T = 9 tensor multiplets,
a finite number of distinct gauge groups and matter content are possible. We
find infinite families of field combinations satisfying anomaly cancellation
and admitting physical gauge kinetic terms for T > 8. We find an integral
lattice associated with each apparently-consistent supergravity theory; this
lattice is determined by the form of the anomaly polynomial. For models which
can be realized in F-theory, this anomaly lattice is related to the
intersection form on the base of the F-theory elliptic fibration. The condition
that a supergravity model have an F-theory realization imposes constraints
which can be expressed in terms of this lattice. The analysis of models which
satisfy known low-energy consistency conditions and yet violate F-theory
constraints suggests possible novel constraints on low-energy supergravity
theories.Comment: 41 pages, 1 figur
Algebraic structural analysis of a vehicle routing problem of heterogeneous trucks. Identification of the properties allowing an exact approach.
Although integer linear programming problems are typically difficult to solve, there exist some easier problems, where the linear programming relaxation is integer.
This thesis sheds light on a drayage problem which is supposed to have this nice feature, after extensive computational experiments.
This thesis aims to provide a theoretical understanding of these results by the analysis of the algebraic structures of the mathematical formulation.
Three reformulations are presented to prove if the constraint matrix is totally unimodular. We will show which experimental conditions are necessary and sufficient (or only sufficient or only necessary) for total unimodularity
Quantization of anomaly coefficients in 6D supergravity
We obtain new constraints on the anomaly coefficients of 6D
supergravity theories using local and global anomaly
cancellation conditions. We show how these constraints can be strengthened if
we assume that the theory is well-defined on any spin space-time with an
arbitrary gauge bundle. We distinguish the constraints depending on the gauge
algebra only from those depending on the global structure of the gauge group.
Our main constraint states that the coefficients of the anomaly polynomial for
the gauge group should be an element of where is the unimodular string charge lattice. We show
that the constraints in their strongest form are realized in F-theory
compactifications. In the process, we identify the cocharacter lattice, which
determines the global structure of the gauge group, within the homology lattice
of the compactification manifold.Comment: 42 pages. v3: Some clarifications, typos correcte
Constraints on 6D Supergravity Theories with Abelian Gauge Symmetry
We study six-dimensional N=(1,0) supergravity theories with abelian, as well
as non-abelian, gauge group factors. We show that for theories with fewer than
nine tensor multiplets, the number of possible combinations of gauge groups -
including abelian factors - and non-abelian matter representations is finite.
We also identify infinite families of theories with distinct U(1) charges that
cannot be ruled out using known quantum consistency conditions, though only a
finite subset of these can arise from known string constructions.Comment: 49 pages, latex; v2: minor corrections, references added; v3: minor
correction
Round and Bipartize for Vertex Cover Approximation
The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a 2-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a round-and-bipartize algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair (?, S), consisting of a graph with an odd cycle transversal.
If S is a stable set, we prove a tight approximation ratio of 1 + 1/?, where 2? -1 denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph ?? : = ?/S and satisfies ? ? [2,?], with ? = ? corresponding to the bipartite case. If S is an arbitrary set, we prove a tight approximation ratio of (1+1/?) (1 - ?) + 2 ?, where ? ? [0,1] is a natural parameter measuring the quality of the set S. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph ??, in combination with an understanding of the weight space where the fully half-integral solution is optimal. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we also obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals, connecting to the MinUncut and Colouring problems. Finally, we show that our analysis is optimal in the following sense: the worst case bounds for ? and ?, which are ? = 2 and ? = 1 - 4/n, recover the integrality gap of 2 - 2/n of the standard linear programming relaxation, where n is the number of vertices of the graph
Global Optimization for Cardinality-constrained Minimum Sum-of-Squares Clustering via Semidefinite Programming
The minimum sum-of-squares clustering (MSSC), or k-means type clustering, has
been recently extended to exploit prior knowledge on the cardinality of each
cluster. Such knowledge is used to increase performance as well as solution
quality. In this paper, we propose a global optimization approach based on the
branch-and-cut technique to solve the cardinality-constrained MSSC. For the
lower bound routine, we use the semidefinite programming (SDP) relaxation
recently proposed by Rujeerapaiboon et al. [SIAM J. Optim. 29(2), 1211-1239,
(2019)]. However, this relaxation can be used in a branch-and-cut method only
for small-size instances. Therefore, we derive a new SDP relaxation that scales
better with the instance size and the number of clusters. In both cases, we
strengthen the bound by adding polyhedral cuts. Benefiting from a tailored
branching strategy which enforces pairwise constraints, we reduce the
complexity of the problems arising in the children nodes. For the upper bound,
instead, we present a local search procedure that exploits the solution of the
SDP relaxation solved at each node. Computational results show that the
proposed algorithm globally solves, for the first time, real-world instances of
size 10 times larger than those solved by state-of-the-art exact methods
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