143 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
University of Windsor Graduate Calendar 2023 Spring
https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1027/thumbnail.jp
University of Windsor Graduate Calendar 2023 Winter
https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1026/thumbnail.jp
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Operational Research: methods and applications
This is the final version. Available on open access from Taylor & Francis via the DOI in this recordThroughout its history, Operational Research has evolved to include methods, models and algorithms that have been applied to a wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first summarises the up-to-date knowledge and provides an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion and used as a point of reference by a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes
The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers
The hypersimplex is the image of the positive Grassmannian
under the moment map. It is a polytope of dimension
in . Meanwhile, the amplituhedron is the
projection of the positive Grassmannian into
under a map induced by a matrix .
Introduced in the context of scattering amplitudes, it is not a polytope, and
has dimension . Nevertheless, there seem to be remarkable connections
between these two objects via T-duality, as was first noted by
Lukowski--Parisi--Williams (LPW). In this paper we use ideas from oriented
matroid theory, total positivity, and the geometry of the hypersimplex and
positroid polytopes to obtain a deeper understanding of the amplituhedron. We
show that the inequalities cutting out positroid polytopes -- images of
positroid cells of under the moment map -- translate into
sign conditions characterizing the T-dual Grasstopes -- images of positroid
cells of under . Moreover, we subdivide the
amplituhedron into chambers, just as the hypersimplex can be subdivided into
simplices, with both chambers and simplices enumerated by the Eulerian numbers.
We prove the main conjecture of (LPW): a collection of positroid polytopes is a
triangulation of if and only if the collection of T-dual
Grasstopes is a triangulation of for all .
Moreover, we prove Arkani-Hamed--Thomas--Trnka's conjectural sign-flip
characterization of , and
Lukowski--Parisi--Spradlin--Volovich's conjectures on cluster adjacency
and on generalized triangles (images of -dimensional positroid cells which
map injectively into ). Finally, we introduce new
cluster structures in the amplituhedron.Comment: 72 pages, many figures, comments welcome. v4: Minor edits v3:
Strengthened results on triangulations and realizability of amplituhedron
sign chambers. v2: Results added to Section 11.4, minor edit
Topology and monoid representations
The goal of this paper is to use topological methods to compute
between an irreducible representation of a finite monoid
inflated from its group completion and one inflated from its group of units, or
more generally coinduced from a maximal subgroup, via a spectral sequence that
collapses on the -page over fields of good characteristic. For von Neumann
regular monoids in which Green's - and -relations
coincide (e.g., left regular bands), the computation of between
arbitrary simple modules reduces to this case, and so our results subsume those
of S. Margolis, F. Saliola, and B. Steinberg, Combinatorial topology and the
global dimension of algebras arising in combinatorics, J. Eur. Math. Soc.
(JEMS), 17, 3037-3080 (2015).
Applications include computing between arbitrary simple
modules and computing a quiver presentation for the algebra of Hsiao's monoid
of ordered -partitions (connected to the Mantaci-Reutenauer descent algebra
for the wreath product ). We show that this algebra is Koszul,
compute its Koszul dual and compute minimal projective resolutions of all the
simple modules using topology. These generalize the results of S. Margolis, F.
V. Saliola, and B. Steinberg. Cell complexes, poset topology and the
representation theory of algebras arising in algebraic combinatorics and
discrete geometry, Mem. Amer. Math. Soc., 274, 1-135, (2021). We also determine
the global dimension of the algebra of the monoid of all affine transformations
of a vector space over a finite field. We provide a topological
characterization of when a monoid homomorphism induces a homological
epimorphism of monoid algebras and apply it to semidirect products. Topology is
used to construct projective resolutions of modules inflated from the group
completion for sufficiently nice monoids
Operational research:methods and applications
Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order
Determinantal Sieving
We introduce determinantal sieving, a new, remarkably powerful tool in the
toolbox of algebraic FPT algorithms. Given a polynomial on a set of
variables and a linear matroid of
rank , both over a field of characteristic 2, in
evaluations we can sieve for those terms in the monomial expansion of which
are multilinear and whose support is a basis for . Alternatively, using
evaluations of we can sieve for those monomials whose odd support
spans . Applying this framework, we improve on a range of algebraic FPT
algorithms, such as:
1. Solving -Matroid Intersection in time and -Matroid
Parity in time , improving on (Brand and Pratt,
ICALP 2021)
2. -Cycle, Colourful -Path, Colourful -Linkage in undirected
graphs, and the more general Rank -Linkage problem, all in
time, improving on respectively (Fomin et al., SODA 2023)
3. Many instances of the Diverse X paradigm, finding a collection of
solutions to a problem with a minimum mutual distance of in time
, improving solutions for -Distinct Branchings from time
to (Bang-Jensen et al., ESA 2021), and for Diverse
Perfect Matchings from to (Fomin et al.,
STACS 2021)
All matroids are assumed to be represented over a field of characteristic 2.
Over general fields, we achieve similar results at the cost of using
exponential space by working over the exterior algebra. For a class of
arithmetic circuits we call strongly monotone, this is even achieved without
any loss of running time. However, the odd support sieving result appears to be
specific to working over characteristic 2
- …