24,228 research outputs found
SPQR-tree-like embedding representation for level planarity
An SPQR-tree is a data structure that efficiently represents all planar embeddings of a connected planar graph. It is a key tool in a number of constrained planarity testing algorithms, which seek a planar embedding of a graph subject to some given set of constraints. We develop an SPQR-tree-like data structure that represents all level-planar embeddings of a biconnected level graph with a single source, called the LP-tree, and give an algorithm to compute it in linear time. Moreover, we show that LP-trees can be used to adapt three constrained planarity algorithms to the level-planar case by using LP-trees as a drop-in replacement for SPQR-trees
An SPQR-Tree-Like Embedding Representation for Level Planarity
An SPQR-tree is a data structure that efficiently represents all planar embeddings of a biconnected planar graph. It is a key tool in a number of constrained planarity testing algorithms, which seek a planar embedding of a graph subject to some given set of constraints.
We develop an SPQR-tree-like data structure that represents all level-planar embeddings of a biconnected level graph with a single source, called the LP-tree, and give a simple algorithm to compute it in linear time. Moreover, we show that LP-trees can be used to adapt three constrained planarity algorithms to the level-planar case by using them as a drop-in replacement for SPQR-trees
An SPQR-tree-like embedding representation for level planarity
An SPQR-tree is a data structure that efficiently represents all planar embeddings of a biconnected planar graph. It is a key tool in a number of constrained planarity testing algorithms, which seek a planar embedding of a graph subject to some given set of constraints. We develop an SPQR-tree-like data structure that represents all level-planar embeddings of a biconnected level graph with a single source, called the LP-tree, and give a simple algorithm to compute it in linear time. Moreover, we show that LP-trees can be used to adapt three constrained planarity algorithms to the level-planar case by using them as a drop-in replacement for SPQR-trees
Learning optimization models in the presence of unknown relations
In a sequential auction with multiple bidding agents, it is highly
challenging to determine the ordering of the items to sell in order to maximize
the revenue due to the fact that the autonomy and private information of the
agents heavily influence the outcome of the auction.
The main contribution of this paper is two-fold. First, we demonstrate how to
apply machine learning techniques to solve the optimal ordering problem in
sequential auctions. We learn regression models from historical auctions, which
are subsequently used to predict the expected value of orderings for new
auctions. Given the learned models, we propose two types of optimization
methods: a black-box best-first search approach, and a novel white-box approach
that maps learned models to integer linear programs (ILP) which can then be
solved by any ILP-solver. Although the studied auction design problem is hard,
our proposed optimization methods obtain good orderings with high revenues.
Our second main contribution is the insight that the internal structure of
regression models can be efficiently evaluated inside an ILP solver for
optimization purposes. To this end, we provide efficient encodings of
regression trees and linear regression models as ILP constraints. This new way
of using learned models for optimization is promising. As the experimental
results show, it significantly outperforms the black-box best-first search in
nearly all settings.Comment: 37 pages. Working pape
Matroidal Degree-Bounded Minimum Spanning Trees
We consider the minimum spanning tree (MST) problem under the restriction
that for every vertex v, the edges of the tree that are adjacent to v satisfy a
given family of constraints. A famous example thereof is the classical
degree-constrained MST problem, where for every vertex v, a simple upper bound
on the degree is imposed. Iterative rounding/relaxation algorithms became the
tool of choice for degree-bounded network design problems. A cornerstone for
this development was the work of Singh and Lau, who showed for the
degree-bounded MST problem how to find a spanning tree violating each degree
bound by at most one unit and with cost at most the cost of an optimal solution
that respects the degree bounds.
However, current iterative rounding approaches face several limits when
dealing with more general degree constraints. In particular, when several
constraints are imposed on the edges adjacent to a vertex v, as for example
when a partition of the edges adjacent to v is given and only a fixed number of
elements can be chosen out of each set of the partition, current approaches
might violate each of the constraints by a constant, instead of violating all
constraints together by at most a constant number of edges. Furthermore, it is
also not clear how previous iterative rounding approaches can be used for
degree constraints where some edges are in a super-constant number of
constraints.
We extend iterative rounding/relaxation approaches both on a conceptual level
as well as aspects involving their analysis to address these limitations. This
leads to an efficient algorithm for the degree-constrained MST problem where
for every vertex v, the edges adjacent to v have to be independent in a given
matroid. The algorithm returns a spanning tree T of cost at most OPT, such that
for every vertex v, it suffices to remove at most 8 edges from T to satisfy the
matroidal degree constraint at v
Improved Algorithm for Degree Bounded Survivable Network Design Problem
We consider the Degree-Bounded Survivable Network Design Problem: the
objective is to find a minimum cost subgraph satisfying the given connectivity
requirements as well as the degree bounds on the vertices. If we denote the
upper bound on the degree of a vertex v by b(v), then we present an algorithm
that finds a solution whose cost is at most twice the cost of the optimal
solution while the degree of a degree constrained vertex v is at most 2b(v) +
2. This improves upon the results of Lau and Singh and that of Lau, Naor,
Salavatipour and Singh
On Generalizations of Network Design Problems with Degree Bounds
Iterative rounding and relaxation have arguably become the method of choice
in dealing with unconstrained and constrained network design problems. In this
paper we extend the scope of the iterative relaxation method in two directions:
(1) by handling more complex degree constraints in the minimum spanning tree
problem (namely, laminar crossing spanning tree), and (2) by incorporating
`degree bounds' in other combinatorial optimization problems such as matroid
intersection and lattice polyhedra. We give new or improved approximation
algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure
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