10 research outputs found
On Graph Crossing Number and Edge Planarization
Given an n-vertex graph G, a drawing of G in the plane is a mapping of its
vertices into points of the plane, and its edges into continuous curves,
connecting the images of their endpoints. A crossing in such a drawing is a
point where two such curves intersect. In the Minimum Crossing Number problem,
the goal is to find a drawing of G with minimum number of crossings. The value
of the optimal solution, denoted by OPT, is called the graph's crossing number.
This is a very basic problem in topological graph theory, that has received a
significant amount of attention, but is still poorly understood
algorithmically. The best currently known efficient algorithm produces drawings
with crossings on bounded-degree graphs, while only a
constant factor hardness of approximation is known. A closely related problem
is Minimum Edge Planarization, in which the goal is to remove a
minimum-cardinality subset of edges from G, such that the remaining graph is
planar. Our main technical result establishes the following connection between
the two problems: if we are given a solution of cost k to the Minimum Edge
Planarization problem on graph G, then we can efficiently find a drawing of G
with at most \poly(d)\cdot k\cdot (k+OPT) crossings, where is the maximum
degree in G. This result implies an O(n\cdot \poly(d)\cdot
\log^{3/2}n)-approximation for Minimum Crossing Number, as well as improved
algorithms for special cases of the problem, such as, for example, k-apex and
bounded-genus graphs
The Effect of Planarization on Width
We study the effects of planarization (the construction of a planar diagram
from a non-planar graph by replacing each crossing by a new vertex) on
graph width parameters. We show that for treewidth, pathwidth, branchwidth,
clique-width, and tree-depth there exists a family of -vertex graphs with
bounded parameter value, all of whose planarizations have parameter value
. However, for bandwidth, cutwidth, and carving width, every graph
with bounded parameter value has a planarization of linear size whose parameter
value remains bounded. The same is true for the treewidth, pathwidth, and
branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Magic-State Functional Units: Mapping and Scheduling Multi-Level Distillation Circuits for Fault-Tolerant Quantum Architectures
Quantum computers have recently made great strides and are on a long-term
path towards useful fault-tolerant computation. A dominant overhead in
fault-tolerant quantum computation is the production of high-fidelity encoded
qubits, called magic states, which enable reliable error-corrected computation.
We present the first detailed designs of hardware functional units that
implement space-time optimized magic-state factories for surface code
error-corrected machines. Interactions among distant qubits require surface
code braids (physical pathways on chip) which must be routed. Magic-state
factories are circuits comprised of a complex set of braids that is more
difficult to route than quantum circuits considered in previous work [1]. This
paper explores the impact of scheduling techniques, such as gate reordering and
qubit renaming, and we propose two novel mapping techniques: braid repulsion
and dipole moment braid rotation. We combine these techniques with graph
partitioning and community detection algorithms, and further introduce a
stitching algorithm for mapping subgraphs onto a physical machine. Our results
show a factor of 5.64 reduction in space-time volume compared to the best-known
previous designs for magic-state factories.Comment: 13 pages, 10 figure
String graphs and separators
String graphs, that is, intersection graphs of curves in the plane, have been
studied since the 1960s. We provide an expository presentation of several
results, including very recent ones: some string graphs require an exponential
number of crossings in every string representation; exponential number is
always sufficient; string graphs have small separators; and the current best
bound on the crossing number of a graph in terms of the pair-crossing number.
For the existence of small separators, unwrapping the complete proof include
generally useful results on approximate flow-cut dualities.Comment: Expository paper based on course note
An Algorithm for the Graph Crossing Number Problem
We study the Minimum Crossing Number problem: given an -vertex graph ,
the goal is to find a drawing of in the plane with minimum number of edge
crossings. This is one of the central problems in topological graph theory,
that has been studied extensively over the past three decades. The first
non-trivial efficient algorithm for the problem, due to Leighton and Rao,
achieved an -approximation for bounded degree graphs. This
algorithm has since been improved by poly-logarithmic factors, with the best
current approximation ratio standing on O(n \poly(d) \log^{3/2}n) for graphs
with maximum degree . In contrast, only APX-hardness is known on the
negative side.
In this paper we present an efficient randomized algorithm to find a drawing
of any -vertex graph in the plane with O(OPT^{10}\cdot \poly(d \log
n)) crossings, where is the number of crossings in the optimal solution,
and is the maximum vertex degree in . This result implies an
\tilde{O}(n^{9/10} \poly(d))-approximation for Minimum Crossing Number, thus
breaking the long-standing -approximation barrier for
bounded-degree graphs
On Graph Crossing Number and Edge Planarization
Given an n-vertex graph G, a drawing of G in the plane is a mapping of its vertices into points of the plane, and its edges into continuous curves, connecting the images of their endpoints. A crossing in such a drawing is a point where two such curves intersect. In the Minimum Crossing Number problem, the goal is to find a drawing of G with minimum number of crossings. The value of the optimal solution, denoted by OPT, is called the graph’s crossing number. This is a very basic problem in topological graph theory, that has received a significant amount of attention, but is still poorly understood algorithmically. The best currently known efficient algorithm produces drawings with O(log 2 n)· (n + OPT) crossings on bounded-degree graphs, while only a constant factor hardness of approximation is known. A closely related problem is Minimum Planarization, in which the goal is to remove a minimum-cardinality subset of edges from G, such that the remaining graph is planar. Our main technical result establishes the following connection between the two problems: if we are given a solution of cost k to the Minimum Planarization problem on graph G, then we can efficiently find a drawing of G with at most poly(d) · k · (k + OPT) crossings, where d is the maximum degree in G. This result implies an O(n · poly(d) · log 3/2 n)approximation for Minimum Crossing Number, as well as improved algorithms for special cases of the problem, such as, for example, k-apex and bounded-genus graphs.