749 research outputs found
Fractional Perfect b-Matching Polytopes. I: General Theory
The fractional perfect b-matching polytope of an undirected graph G is the
polytope of all assignments of nonnegative real numbers to the edges of G such
that the sum of the numbers over all edges incident to any vertex v is a
prescribed nonnegative number b_v. General theorems which provide conditions
for nonemptiness, give a formula for the dimension, and characterize the
vertices, edges and face lattices of such polytopes are obtained. Many of these
results are expressed in terms of certain spanning subgraphs of G which are
associated with subsets or elements of the polytope. For example, it is shown
that an element u of the fractional perfect b-matching polytope of G is a
vertex of the polytope if and only if each component of the graph of u either
is acyclic or else contains exactly one cycle with that cycle having odd
length, where the graph of u is defined to be the spanning subgraph of G whose
edges are those at which u is positive.Comment: 37 page
The weighted stable matching problem
We study the stable matching problem in non-bipartite graphs with incomplete
but strict preference lists, where the edges have weights and the goal is to
compute a stable matching of minimum or maximum weight. This problem is known
to be NP-hard in general. Our contribution is two fold: a polyhedral
characterization and an approximation algorithm. Previously Chen et al. have
shown that the stable matching polytope is integral if and only if the subgraph
obtained after running phase one of Irving's algorithm is bipartite. We improve
upon this result by showing that there are instances where this subgraph might
not be bipartite but one can further eliminate some edges and arrive at a
bipartite subgraph. Our elimination procedure ensures that the set of stable
matchings remains the same, and thus the stable matching polytope of the final
subgraph contains the incidence vectors of all stable matchings of our original
graph. This allows us to characterize a larger class of instances for which the
weighted stable matching problem is polynomial-time solvable. We also show that
our edge elimination procedure is best possible, meaning that if the subgraph
we arrive at is not bipartite, then there is no bipartite subgraph that has the
same set of stable matchings as the original graph. We complement these results
with a -approximation algorithm for the minimum weight stable matching
problem for instances where each agent has at most two possible partners in any
stable matching. This is the first approximation result for any class of
instances with general weights.Comment: This is an extended version of a paper to appear at the The Fourth
International Workshop on Matching Under Preferences (MATCH-UP 2017
Faces of Birkhoff Polytopes
The Birkhoff polytope B(n) is the convex hull of all (n x n) permutation
matrices, i.e., matrices where precisely one entry in each row and column is
one, and zeros at all other places. This is a widely studied polytope with
various applications throughout mathematics.
In this paper we study combinatorial types L of faces of a Birkhoff polytope.
The Birkhoff dimension bd(L) of L is the smallest n such that B(n) has a face
with combinatorial type L.
By a result of Billera and Sarangarajan, a combinatorial type L of a
d-dimensional face appears in some B(k) for k less or equal to 2d, so bd(L) is
at most d. We will characterize those types whose Birkhoff dimension is at
least 2d-3, and we prove that any type whose Birkhoff dimension is at least d
is either a product or a wedge over some lower dimensional face. Further, we
computationally classify all d-dimensional combinatorial types for d between 2
and 8.Comment: 29 page
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