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Weighted Well-Covered Claw-Free Graphs
A graph G is well-covered if all its maximal independent sets are of the same
cardinality. Assume that a weight function w is defined on its vertices. Then G
is w-well-covered if all maximal independent sets are of the same weight. For
every graph G, the set of weight functions w such that G is w-well-covered is a
vector space. Given an input claw-free graph G, we present an O(n^6)algortihm,
whose input is a claw-free graph G, and output is the vector space of weight
functions w, for which G is w-well-covered. A graph G is equimatchable if all
its maximal matchings are of the same cardinality. Assume that a weight
function w is defined on the edges of G. Then G is w-equimatchable if all its
maximal matchings are of the same weight. For every graph G, the set of weight
functions w such that G is w-equimatchable is a vector space. We present an
O(m*n^4 + n^5*log(n)) algorithm which receives an input graph G, and outputs
the vector space of weight functions w such that G is w-equimatchable.Comment: 14 pages, 1 figur
Quantum Hamiltonian reduction of W-algebras and category O
W-algebras are a class of non-commutative algebras related to the classical
universal enveloping algebras. They can be defined as a subquotient of U(g)
related to a choice of nilpotent element e and compatible nilpotent subalgebra
m. The definition is a quantum analogue of the classical construction of
Hamiltonian reduction.
We define a quantum version of Hamiltonian reduction by stages and use it to
construct intermediate reductions between different W-algebras U(g,e) in type
A.This allows us to express the W-algebra U(g,e') as a subquotient of U(g,e)
for nilpotent elements e' covering e. It also produces a collection of
(U(g,e),U(g,e'))-bimodules analogous to the generalised Gel'fand-Graev modules
used in the classical definition of the W-algebra; these can be used to obtain
adjoint functors between the corresponding module categories.
The category of modules over a W-algebra has a full subcategory defined in a
parallel fashion to that of the Bernstein-Gel'fand-Gel'fand (BGG) category O;
this version of category O(e) for W-algebras is equivalent to an infinitesimal
block of O by an argument of Mili\v{c}i\'{c} and Soergel. We therefore
construct analogues of the translation functors between the different blocks of
O, in this case being functors between the categories O(e) for different
W-algebras U(g,e). This follows an argument of Losev, and realises the category
O(e') as equivalent to a full subcategory of the category O(e) where e' is
greater than e in the refinement ordering.Comment: University of Toronto PhD thesis, defended July 2014, 57 page
A Decomposition Theorem for Maximum Weight Bipartite Matchings
Let G be a bipartite graph with positive integer weights on the edges and
without isolated nodes. Let n, N and W be the node count, the largest edge
weight and the total weight of G. Let k(x,y) be log(x)/log(x^2/y). We present a
new decomposition theorem for maximum weight bipartite matchings and use it to
design an O(sqrt(n)W/k(n,W/N))-time algorithm for computing a maximum weight
matching of G. This algorithm bridges a long-standing gap between the best
known time complexity of computing a maximum weight matching and that of
computing a maximum cardinality matching. Given G and a maximum weight matching
of G, we can further compute the weight of a maximum weight matching of G-{u}
for all nodes u in O(W) time.Comment: The journal version will appear in SIAM Journal on Computing. The
conference version appeared in ESA 199
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