784 research outputs found
Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut
We investigate the notion of stability proposed by Bilu and Linial. We obtain
an exact polynomial-time algorithm for -stable Max Cut instances with
for some absolute constant . Our
algorithm is robust: it never returns an incorrect answer; if the instance is
-stable, it finds the maximum cut, otherwise, it either finds the
maximum cut or certifies that the instance is not -stable. We prove
that there is no robust polynomial-time algorithm for -stable instances
of Max Cut when , where is the best
approximation factor for Sparsest Cut with non-uniform demands.
Our algorithm is based on semidefinite programming. We show that the standard
SDP relaxation for Max Cut (with triangle inequalities) is integral
if , where
is the least distortion with which every point metric space of negative
type embeds into . On the negative side, we show that the SDP
relaxation is not integral when .
Moreover, there is no tractable convex relaxation for -stable instances
of Max Cut when . That suggests that solving
-stable instances with might be difficult or
impossible.
Our results significantly improve previously known results. The best
previously known algorithm for -stable instances of Max Cut required
that (for some ) [Bilu, Daniely, Linial, and
Saks]. No hardness results were known for the problem. Additionally, we present
an algorithm for 4-stable instances of Minimum Multiway Cut. We also study a
relaxed notion of weak stability.Comment: 24 page
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A new partitioning approach for layout synthesis from register-transfer netlists
Most of the IC today are described and documented using heiarchical netlists. In addition to gates, latches, and flip-flops, these netlists include sliceable register-transfer components such as registers, counters, adders, ALUs, shifters, register files, and multiplexers. Usually, these components are decomposed into basic gates, latches, and flip-flops, and are laid out using standard cells. The standard cell architecture requires excessive routing area, and does not exploit the bit-sliced nature of register-transfer components. In this paper, we present a new sliced-layout architecture to alleviate the preceding problems. We also describe partitioning algorithms that are used to generate the floorplan for this layout architecture. The partitioning algorithms not only select the best suited layout style for each component, but also consider critical paths, I/O pin locations, and connections between blocks. This approach improves the overall area utilization and minimizes the total wire length
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
Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset
Given a directed graph , a set of terminals and an integer , the
\textsc{Directed Vertex Multiway Cut} problem asks if there is a set of at
most (nonterminal) vertices whose removal disconnects each terminal from
all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous
problem where is a set of at most edges. These two problems indeed are
known to be equivalent. A natural generalization of the multiway cut is the
\emph{multicut} problem, in which we want to disconnect only a set of given
pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in
undirected graphs multiway cut is fixed-parameter tractable (FPT) parameterized
by . Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and
directed multicut is W[1]-hard parameterized by . We complete the picture
here by our main result which is that both \textsc{Directed Vertex Multiway
Cut} and \textsc{Directed Edge Multiway Cut} can be solved in time
, i.e., FPT parameterized by size of the cutset of
the solution. This answers an open question raised by Marx (Theor. Comp. Sci.
2006) and Marx and Razgon (STOC 2011). It follows from our result that
\textsc{Directed Multicut} is FPT for the case of terminal pairs, which
answers another open problem raised in Marx and Razgon (STOC 2011)
Fixed-parameter tractability of multicut parameterized by the size of the cutset
Given an undirected graph , a collection of
pairs of vertices, and an integer , the Edge Multicut problem ask if there
is a set of at most edges such that the removal of disconnects
every from the corresponding . Vertex Multicut is the analogous
problem where is a set of at most vertices. Our main result is that
both problems can be solved in time , i.e.,
fixed-parameter tractable parameterized by the size of the cutset in the
solution. By contrast, it is unlikely that an algorithm with running time of
the form exists for the directed version of the problem, as
we show it to be W[1]-hard parameterized by the size of the cutset
Multiway Spectral Clustering: A Margin-Based Perspective
Spectral clustering is a broad class of clustering procedures in which an
intractable combinatorial optimization formulation of clustering is "relaxed"
into a tractable eigenvector problem, and in which the relaxed solution is
subsequently "rounded" into an approximate discrete solution to the original
problem. In this paper we present a novel margin-based perspective on multiway
spectral clustering. We show that the margin-based perspective illuminates both
the relaxation and rounding aspects of spectral clustering, providing a unified
analysis of existing algorithms and guiding the design of new algorithms. We
also present connections between spectral clustering and several other topics
in statistics, specifically minimum-variance clustering, Procrustes analysis
and Gaussian intrinsic autoregression.Comment: Published in at http://dx.doi.org/10.1214/08-STS266 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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