196 research outputs found

### Local Guarantees in Graph Cuts and Clustering

Correlation Clustering is an elegant model that captures fundamental graph
cut problems such as Min $s-t$ 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 $s-t$ Cut: find an $s-t$ cut
minimizing the largest number of cut edges incident on any node. We present the
following results: $(1)$ an $O(\sqrt{n})$-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), $(2)$ a remarkably simple $7$-approximation for minimizing
local disagreements in complete graphs (improving upon the previous best known
approximation of $48$), and $(3)$ a $1/(2+\varepsilon)$-approximation for
maximizing the minimum total weight of agreement edges incident on any node,
hence improving upon the $1/(4+\varepsilon)$-approximation that follows from
the study of approximate pure Nash equilibria in cut and party affiliation
games

### On Deterministic Sketching and Streaming for Sparse Recovery and Norm Estimation

We study classic streaming and sparse recovery problems using deterministic
linear sketches, including l1/l1 and linf/l1 sparse recovery problems (the
latter also being known as l1-heavy hitters), norm estimation, and approximate
inner product. We focus on devising a fixed matrix A in R^{m x n} and a
deterministic recovery/estimation procedure which work for all possible input
vectors simultaneously. Our results improve upon existing work, the following
being our main contributions:
* A proof that linf/l1 sparse recovery and inner product estimation are
equivalent, and that incoherent matrices can be used to solve both problems.
Our upper bound for the number of measurements is m=O(eps^{-2}*min{log n, (log
n / log(1/eps))^2}). We can also obtain fast sketching and recovery algorithms
by making use of the Fast Johnson-Lindenstrauss transform. Both our running
times and number of measurements improve upon previous work. We can also obtain
better error guarantees than previous work in terms of a smaller tail of the
input vector.
* A new lower bound for the number of linear measurements required to solve
l1/l1 sparse recovery. We show Omega(k/eps^2 + klog(n/k)/eps) measurements are
required to recover an x' with |x - x'|_1 <= (1+eps)|x_{tail(k)}|_1, where
x_{tail(k)} is x projected onto all but its largest k coordinates in magnitude.
* A tight bound of m = Theta(eps^{-2}log(eps^2 n)) on the number of
measurements required to solve deterministic norm estimation, i.e., to recover
|x|_2 +/- eps|x|_1.
For all the problems we study, tight bounds are already known for the
randomized complexity from previous work, except in the case of l1/l1 sparse
recovery, where a nearly tight bound is known. Our work thus aims to study the
deterministic complexities of these problems

### On absolute stability of nonlinear systems with small delays

Nonlinear nonautonomous retarded systems with separated autonomous linear parts
and continuous nonlinear ones are considered. It is assumed that deviations of the argument are sufficiently small. Absolute stability conditions are derived. They are formulated in terms of eigenvalues of auxiliary matrices

### Exact Weight Subgraphs and the k-Sum Conjecture

We consider the Exact-Weight-H problem of finding a (not necessarily induced)
subgraph H of weight 0 in an edge-weighted graph G. We show that for every H,
the complexity of this problem is strongly related to that of the infamous
k-Sum problem. In particular, we show that under the k-Sum Conjecture, we can
achieve tight upper and lower bounds for the Exact-Weight-H problem for various
subgraphs H such as matching, star, path, and cycle. One interesting
consequence is that improving on the O(n^3) upper bound for Exact-Weight-4-Path
or Exact-Weight-5-Path will imply improved algorithms for 3-Sum, 5-Sum,
All-Pairs Shortest Paths and other fundamental problems. This is in sharp
contrast to the minimum-weight and (unweighted) detection versions, which can
be solved easily in time O(n^2). We also show that a faster algorithm for any
of the following three problems would yield faster algorithms for the others:
3-Sum, Exact-Weight-3-Matching, and Exact-Weight-3-Star

### Cluster Editing: Kernelization based on Edge Cuts

Kernelization algorithms for the {\sc cluster editing} problem have been a
popular topic in the recent research in parameterized computation. Thus far
most kernelization algorithms for this problem are based on the concept of {\it
critical cliques}. In this paper, we present new observations and new
techniques for the study of kernelization algorithms for the {\sc cluster
editing} problem. Our techniques are based on the study of the relationship
between {\sc cluster editing} and graph edge-cuts. As an application, we
present an ${\cal O}(n^2)$-time algorithm that constructs a $2k$ kernel for the
{\it weighted} version of the {\sc cluster editing} problem. Our result meets
the best kernel size for the unweighted version for the {\sc cluster editing}
problem, and significantly improves the previous best kernel of quadratic size
for the weighted version of the problem

### Rare Complications of Cervical Spine Surgery: Pseudomeningocoele.

STUDY DESIGN: This study was a retrospective, multicenter cohort study.
OBJECTIVES: Rare complications of cervical spine surgery are inherently difficult to investigate. Pseudomeningocoele (PMC), an abnormal collection of cerebrospinal fluid that communicates with the subarachnoid space, is one such complication. In order to evaluate and better understand the incidence, presentation, treatment, and outcome of PMC following cervical spine surgery, we conducted a multicenter study to pool our collective experience.
METHODS: This study was a retrospective, multicenter cohort study of patients who underwent cervical spine surgery at any level(s) from C2 to C7, inclusive; were over 18 years of age; and experienced a postoperative PMC.
RESULTS: Thirteen patients (0.08%) developed a postoperative PMC, 6 (46.2%) of whom were female. They had an average age of 48.2 years and stayed in hospital a mean of 11.2 days. Three patients were current smokers, 3 previous smokers, 5 had never smoked, and 2 had unknown smoking status. The majority, 10 (76.9%), were associated with posterior surgery, whereas 3 (23.1%) occurred after an anterior procedure. Myelopathy was the most common indication for operations that were complicated by PMC (46%). Seven patients (53%) required a surgical procedure to address the PMC, whereas the remaining 6 were treated conservatively. All PMCs ultimately resolved or were successfully treated with no residual effects.
CONCLUSIONS: PMC is a rare complication of cervical surgery with an incidence of less than 0.1%. They prolong hospital stay. PMCs occurred more frequently in association with posterior approaches. Approximately half of PMCs required surgery and all ultimately resolved without residual neurologic or other long-term effects

### User-friendly tail bounds for sums of random matrices

This paper presents new probability inequalities for sums of independent,
random, self-adjoint matrices. These results place simple and easily verifiable
hypotheses on the summands, and they deliver strong conclusions about the
large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for
the norm of a sum of random rectangular matrices follow as an immediate
corollary. The proof techniques also yield some information about matrix-valued
martingales.
In other words, this paper provides noncommutative generalizations of the
classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff,
Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of
application, ease of use, and strength of conclusion that have made the scalar
inequalities so valuable.Comment: Current paper is the version of record. The material on Freedman's
inequality has been moved to a separate note; other martingale bounds are
described in Caltech ACM Report 2011-0

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