644 research outputs found
An SDP approach to multi-level crossing minimization
We present an approach based on semidefinite programs (SDP) to tackle the multi-level crossing minimization prob- lem. Thereby, we are given a layered graph (i.e., the graph´s vertices are assigned to multiple parallel levels) and ask for an ordering of the nodes on their levels such that, when draw- ing the graph with straight lines, the resulting number of crossings is minimized. Solving this step is crucial in the probably most widely used graph drawing scheme, the so- called Sugiyama framework. The problem has received a lot of attention both in the field of heuristics and exact methods. For a long time, integer linear programming (ILP) approaches were the only exact algorithms applicable at least to small graphs. Recently, SDP formulations for the special case of two levels were proposed and dominated the ILP for dense instances. In this paper, we present a new SDP formulation for the general multi-level version that, for two-levels, is even stronger than the aforementioned specialized SDP. As a side- product, we also obtain an SDP-based heuristic which in practice always gives (near-)optimal solutions. We conduct a large set of experiments, both on random- ized and on real-world instances, and compare our approach to a state-of-the-art ILP-based branch-and-cut implementa- tion. The SDP clearly dominates for denser graphs, while the ILP approach is usually faster for sparse instances. However, even for such sparse graphs, the SDP solves more instances to optimality than the ILP. In fact, there is no single instance the ILP solved, which the SDP did not. Overall, our experi- ments reveal that for sparse graphs, one should usually try to find an optimal solution with the ILP first. If this approach does not solve the instance to optimality within reasonable time, the SDP still has a good chance to do so. Being able to solve larger real-world instances than reported before, we are also able to evaluate heuristics for this problem. In this paper we do so for the traditional barycenter-heuristic (showing that it leaves a large gap to the true optimum) and the state-of-the-art upward-planarization method (showing that it is usually close to the optimum)
Local Guarantees in Graph Cuts and Clustering
Correlation Clustering is an elegant model that captures fundamental graph
cut problems such as Min Cut, Multiway Cut, and Multicut, extensively
studied in combinatorial optimization. Here, we are given a graph with edges
labeled or and the goal is to produce a clustering that agrees with the
labels as much as possible: edges within clusters and edges across
clusters. The classical approach towards Correlation Clustering (and other
graph cut problems) is to optimize a global objective. We depart from this and
study local objectives: minimizing the maximum number of disagreements for
edges incident on a single node, and the analogous max min agreements
objective. This naturally gives rise to a family of basic min-max graph cut
problems. A prototypical representative is Min Max Cut: find an cut
minimizing the largest number of cut edges incident on any node. We present the
following results: an -approximation for the problem of
minimizing the maximum total weight of disagreement edges incident on any node
(thus providing the first known approximation for the above family of min-max
graph cut problems), a remarkably simple -approximation for minimizing
local disagreements in complete graphs (improving upon the previous best known
approximation of ), and a -approximation for
maximizing the minimum total weight of agreement edges incident on any node,
hence improving upon the -approximation that follows from
the study of approximate pure Nash equilibria in cut and party affiliation
games
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Layer assignment and routing optimization for advanced technologies
As VLSI technology scales to deep sub-micron and beyond, it becomes
increasingly challenging to achieve timing closure for VLSI design. Since a
complete design flow consists of several phases, such as logic synthesis, placement, and routing, interconnect synthesis plays an important role which includes buffer insertion/sizing and timing-driven routing. Although progress has been achieved by many advanced routing techniques, the following aspects
can be exploited sufficiently for further improvement: (1) incremental layer assignment for timing optimization; (2) signal routing with the requirement of regularity; (3) power-efficient optical-electrical interconnect paradigm. Thus, to perform the layer assignment and routing optimization for advanced technologies,
an automated routing engine in a global view is essential to benefit the interconnect design while satisfying specific requirements.
This dissertation proposes a set of algorithms and methodology on layer
assignment and routing optimization for advanced technologies. The research includes two timing-driven incremental layer assignment approaches, synergistic
topology generation and routing synthesis for signal groups, and optical-electrical routing design for power efficiency.
For incremental layer assignment, most of the conventional approaches
target via minimization but neglect the timing issues. Meanwhile, via delays
are ignored but should be considered in emerging technology nodes. Then two
timing-driven incremental layer assignment frameworks are proposed, where all the nets are solved simultaneously with the integration of via delays: (1) optimization of the total sum of net delays and reduction of slew violations; (2) minimization of critical path timing in selected nets.
For on-chip signal routing, the bundled bits in one group may have different
pin locations, but they have to be routed in a regular manner by sharing common topologies. Very few previous works target inter-bit regularity via multi-layer topology selection. Furthermore, the routability and wire-length of the signal bits should also be optimized. Then an advanced synergistic routing engine is promoted, which is able to not only control routability and wire-length but also guide each bit routing intelligently for design regularity.
For optical-electrical co-design routing, optical interconnect shows its
advantage due to the dominance of bandwidth-distance-power properties. The previous works lack a detailed exploration of optical-electrical co-design for on-chip interconnects. During the transmission, signal quality can be affected by various loss sources and Electrical to Optical (EO)/Optical to Electrical (OE) conversion overheads should also be considered. Then a power-efficient routing flow for on-chip signals is presented, where optical connections can collaborate with electrical wires seamlessly.
The effectiveness of proposed algorithms and techniques is demonstrated in this dissertation. These approaches are able to achieve the improvements regarding specific metrics and eventually benefit the routing flow.Electrical and Computer Engineerin
On a generalization of iterated and randomized rounding
We give a general method for rounding linear programs that combines the
commonly used iterated rounding and randomized rounding techniques. In
particular, we show that whenever iterated rounding can be applied to a problem
with some slack, there is a randomized procedure that returns an integral
solution that satisfies the guarantees of iterated rounding and also has
concentration properties. We use this to give new results for several classic
problems where iterated rounding has been useful
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