713 research outputs found
The Complexity of Distributed Edge Coloring with Small Palettes
The complexity of distributed edge coloring depends heavily on the palette
size as a function of the maximum degree . In this paper we explore the
complexity of edge coloring in the LOCAL model in different palette size
regimes.
1. We simplify the \emph{round elimination} technique of Brandt et al. and
prove that -edge coloring requires
time w.h.p. and time deterministically, even on trees.
The simplified technique is based on two ideas: the notion of an irregular
running time and some general observations that transform weak lower bounds
into stronger ones.
2. We give a randomized edge coloring algorithm that can use palette sizes as
small as , which is a natural barrier for
randomized approaches. The running time of the algorithm is at most
, where is the complexity of a
permissive version of the constructive Lovasz local lemma.
3. We develop a new distributed Lovasz local lemma algorithm for
tree-structured dependency graphs, which leads to a -edge
coloring algorithm for trees running in time. This algorithm
arises from two new results: a deterministic -time LLL algorithm for
tree-structured instances, and a randomized -time graph
shattering method for breaking the dependency graph into independent -size LLL instances.
4. A natural approach to computing -edge colorings (Vizing's
theorem) is to extend partial colorings by iteratively re-coloring parts of the
graph. We prove that this approach may be viable, but in the worst case
requires recoloring subgraphs of diameter . This stands
in contrast to distributed algorithms for Brooks' theorem, which exploit the
existence of -length augmenting paths
Escaping the Local Minima via Simulated Annealing: Optimization of Approximately Convex Functions
We consider the problem of optimizing an approximately convex function over a
bounded convex set in using only function evaluations. The
problem is reduced to sampling from an \emph{approximately} log-concave
distribution using the Hit-and-Run method, which is shown to have the same
complexity as sampling from log-concave distributions. In
addition to extend the analysis for log-concave distributions to approximate
log-concave distributions, the implementation of the 1-dimensional sampler of
the Hit-and-Run walk requires new methods and analysis. The algorithm then is
based on simulated annealing which does not relies on first order conditions
which makes it essentially immune to local minima.
We then apply the method to different motivating problems. In the context of
zeroth order stochastic convex optimization, the proposed method produces an
-minimizer after noisy function
evaluations by inducing a -approximately log concave
distribution. We also consider in detail the case when the "amount of
non-convexity" decays towards the optimum of the function. Other applications
of the method discussed in this work include private computation of empirical
risk minimizers, two-stage stochastic programming, and approximate dynamic
programming for online learning.Comment: 27 page
Graph-theoretical Bounds on the Entangled Value of Non-local Games
We introduce a novel technique to give bounds to the entangled value of
non-local games. The technique is based on a class of graphs used by Cabello,
Severini and Winter in 2010. The upper bound uses the famous Lov\'asz theta
number and is efficiently computable; the lower one is based on the quantum
independence number, which is a quantity used in the study of
entanglement-assisted channel capacities and graph homomorphism games.Comment: 10 pages, submission to the 9th Conference on the Theory of Quantum
Computation, Communication, and Cryptography (TQC 2014
Improved Distributed Algorithms for the Lovász Local Lemma and Edge Coloring
The Lovász Local Lemma is a classic result in probability theory that is often used to prove the existence of combinatorial objects via the probabilistic method. In its simplest form, it states that if we have n ‘bad events’, each of which occurs with probability at most p and is independent of all but d other events, then under certain criteria on p and d, all of the bad events can be avoided with positive probability. While the original proof was existential, there has been much study on the algorithmic Lovász Local Lemma: that is, designing an algorithm which finds an assignment of the underlying random variables such that all the bad events are indeed avoided. Notably, the celebrated result of Moser and Tardos [JACM ’10] also implied an efficient distributed algorithm for the problem, running in O(log2 n) rounds. For instances with low d, this was improved to O(d 2 + logO(1) log n) by Fischer and Ghaffari [DISC ’17], a result that has proven highly important in distributed complexity theory (Chang and Pettie [SICOMP ’19]). We give an improved algorithm for the Lovász Local Lemma, providing a trade-off between the strength of the criterion relating p and d, and the distributed round complexity. In particular, in the same regime as Fischer and Ghaffari’s algorithm, we improve the round complexity to O( d log d + logO(1) log n). At the other end of the trade-off, we obtain a logO(1) log n round complexity for a substantially wider regime than previously known. As our main application, we also give the first logO(1) log n-round distributed algorithm for the problem of ∆+o(∆)-edge coloring a graph of maximum degree ∆. This is an almost exponential improvement over previous results: no prior logo(1) n-round algorithm was known even for 2∆ − 2-edge coloring
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