84 research outputs found
Commutative Algorithms Approximate the LLL-distribution
Following the groundbreaking Moser-Tardos algorithm for the Lovasz Local
Lemma (LLL), a series of works have exploited a key ingredient of the original
analysis, the witness tree lemma, in order to: derive deterministic, parallel
and distributed algorithms for the LLL, to estimate the entropy of the output
distribution, to partially avoid bad events, to deal with super-polynomially
many bad events, and even to devise new algorithmic frameworks. Meanwhile, a
parallel line of work, has established tools for analyzing stochastic local
search algorithms motivated by the LLL that do not fall within the Moser-Tardos
framework. Unfortunately, the aforementioned results do not transfer to these
more general settings. Mainly, this is because the witness tree lemma,
provably, no longer holds. Here we prove that for commutative algorithms, a
class recently introduced by Kolmogorov and which captures the vast majority of
LLL applications, the witness tree lemma does hold. Armed with this fact, we
extend the main result of Haeupler, Saha, and Srinivasan to commutative
algorithms, establishing that the output of such algorithms well-approximates
the LLL-distribution, i.e., the distribution obtained by conditioning on all
bad events being avoided, and give several new applications. For example, we
show that the recent algorithm of Molloy for list coloring number of sparse,
triangle-free graphs can output exponential many list colorings of the input
graph
Simple Local Computation Algorithms for the General Lovasz Local Lemma
We consider the task of designing Local Computation Algorithms (LCA) for
applications of the Lov\'{a}sz Local Lemma (LLL). LCA is a class of sublinear
algorithms proposed by Rubinfeld et al.~\cite{Ronitt} that have received a lot
of attention in recent years. The LLL is an existential, sufficient condition
for a collection of sets to have non-empty intersection (in applications,
often, each set comprises all objects having a certain property). The
ground-breaking algorithm of Moser and Tardos~\cite{MT} made the LLL fully
constructive, following earlier results by Beck~\cite{beck_lll} and
Alon~\cite{alon_lll} giving algorithms under significantly stronger LLL-like
conditions. LCAs under those stronger conditions were given in~\cite{Ronitt},
where it was asked if the Moser-Tardos algorithm can be used to design LCAs
under the standard LLL condition. The main contribution of this paper is to
answer this question affirmatively. In fact, our techniques yield LCAs for
settings beyond the standard LLL condition
Quantum Computation, Markov Chains and Combinatorial Optimisation
This thesis addresses two questions related to the title, Quantum Computation, Markov Chains and Combinatorial Optimisation. The first question involves an algorithmic primitive of quantum computation, quantum walks on graphs, and its relation to Markov Chains. Quantum walks have been shown in certain cases to mix faster than their classical counterparts. Lifted Markov chains, consisting of a Markov chain on an extended state space which is projected back down to the original state space, also show considerable speedups in mixing time. We design a lifted Markov chain that in some sense simulates any quantum walk. Concretely, we construct a lifted Markov chain on a connected graph G with n vertices that mixes exactly to the average mixing distribution of a quantum walk on G. Moreover, the mixing time of this chain is the diameter of G. We then consider practical consequences of this result. In the second part of this thesis we address a classic unsolved problem in combinatorial optimisation, graph isomorphism. A theorem of Kozen states that two graphs on n vertices are isomorphic if and only if there is a clique of size n in the weak modular product of the two graphs. Furthermore, a straightforward corollary of this theorem and LovaÌszâs sandwich theorem is if the weak modular product between two graphs is perfect, then checking if the graphs are isomorphic is polynomial in n. We enumerate the necessary and sufficient conditions for the weak modular product of two simple graphs to be perfect. Interesting cases include complete multipartite graphs and disjoint unions of cliques. We find that all perfect weak modular products have factors that fall into classes of graphs for which testing isomorphism is already known to be polynomial in the number of vertices. Open questions and further research directions are discussed
Hidden Variables for Pauli Measurements
The Pauli measurements (the measurements that can be performed with Clifford
operators followed by measurement in the computational basis) are a fundamental
object in quantum information. It is well-known that there is no assignment of
outcomes to all Pauli measurements that is both complete and consistent. We
define two classes of hidden variable assignments based on relaxing either
condition. Partial hidden variable assignments retain the consistency
condition, but forfeit completeness. Contextual hidden variable assignments
retain completeness but forfeit consistency. We use techniques from spectral
graph theory to characterize the incompleteness and inconsistency of the
respective hidden variable assignments. As an application, we interpret our
incompleteness result as a statement of contextuality and our inconsistency
result as a statement of nonlocality. Our results show that we can obtain large
amounts of contextuality and nonlocality using Clifford gates and measurements
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