7,024 research outputs found
Robust Assignments via Ear Decompositions and Randomized Rounding
Many real-life planning problems require making a priori decisions before all
parameters of the problem have been revealed. An important special case of such
problem arises in scheduling problems, where a set of tasks needs to be
assigned to the available set of machines or personnel (resources), in a way
that all tasks have assigned resources, and no two tasks share the same
resource. In its nominal form, the resulting computational problem becomes the
\emph{assignment problem} on general bipartite graphs.
This paper deals with a robust variant of the assignment problem modeling
situations where certain edges in the corresponding graph are \emph{vulnerable}
and may become unavailable after a solution has been chosen. The goal is to
choose a minimum-cost collection of edges such that if any vulnerable edge
becomes unavailable, the remaining part of the solution contains an assignment
of all tasks.
We present approximation results and hardness proofs for this type of
problems, and establish several connections to well-known concepts from
matching theory, robust optimization and LP-based techniques.Comment: Full version of ICALP 2016 pape
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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