4,993 research outputs found
Gap Amplification for Small-Set Expansion via Random Walks
In this work, we achieve gap amplification for the Small-Set Expansion
problem. Specifically, we show that an instance of the Small-Set Expansion
Problem with completeness and soundness is at least as
difficult as Small-Set Expansion with completeness and soundness
, for any function which grows faster than
. We achieve this amplification via random walks -- our gadget
is the graph with adjacency matrix corresponding to a random walk on the
original graph. An interesting feature of our reduction is that unlike gap
amplification via parallel repetition, the size of the instances (number of
vertices) produced by the reduction remains the same
The Quantum PCP Conjecture
The classical PCP theorem is arguably the most important achievement of
classical complexity theory in the past quarter century. In recent years,
researchers in quantum computational complexity have tried to identify
approaches and develop tools that address the question: does a quantum version
of the PCP theorem hold? The story of this study starts with classical
complexity and takes unexpected turns providing fascinating vistas on the
foundations of quantum mechanics, the global nature of entanglement and its
topological properties, quantum error correction, information theory, and much
more; it raises questions that touch upon some of the most fundamental issues
at the heart of our understanding of quantum mechanics. At this point, the jury
is still out as to whether or not such a theorem holds. This survey aims to
provide a snapshot of the status in this ongoing story, tailored to a general
theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column
from Volume 44 Issue 2, June 201
Finding a marked node on any graph by continuous-time quantum walk
Spatial search by discrete-time quantum walk can find a marked node on any
ergodic, reversible Markov chain quadratically faster than its classical
counterpart, i.e.\ in a time that is in the square root of the hitting time of
. However, in the framework of continuous-time quantum walks, it was
previously unknown whether such general speed-up is possible. In fact, in this
framework, the widely used quantum algorithm by Childs and Goldstone fails to
achieve such a speedup. Furthermore, it is not clear how to apply this
algorithm for searching any Markov chain . In this article, we aim to
reconcile the apparent differences between the running times of spatial search
algorithms in these two frameworks. We first present a modified version of the
Childs and Goldstone algorithm which can search for a marked element for any
ergodic, reversible by performing a quantum walk on its edges. Although
this approach improves the algorithmic running time for several instances, it
cannot provide a generic quadratic speedup for any . Secondly, using the
framework of interpolated Markov chains, we provide a new spatial search
algorithm by continuous-time quantum walk which can find a marked node on any
in the square root of the classical hitting time. In the scenario where
multiple nodes are marked, the algorithmic running time scales as the square
root of a quantity known as the extended hitting time. Our results establish a
novel connection between discrete-time and continuous-time quantum walks and
can be used to develop a number of Markov chain-based quantum algorithms.Comment: This version deals only with new algorithms for spatial search by
continuous-time quantum walk (CTQW) on ergodic, reversible Markov chains.
Please see arXiv:2004.12686 for results on the necessary and sufficient
conditions for the optimality of the Childs and Goldstone algorithm for
spatial search by CTQ
Monotone expansion
This work, following the outline set in [B2], presents an explicit
construction of a family of monotone expanders. The family is essentially
defined by the Mobius action of SL_2(R) on the real line. For the proof, we
show a product-growth theorem for SL_2(R).Comment: 37 page
Quantum fast-forwarding: Markov chains and graph property testing
We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with P the Markov chain transition matrix and D=P∘PT its discriminant matrix (D=P if P is symmetric), we construct a quantum walk algorithm that for any quantum state |v⟩ and integer t returns a quantum state ϵ-close to the state Dt|v⟩/∥Dt|v⟩∥. The algorithm uses O(∥Dt|v⟩∥−1tlog(ϵ∥Dt|v⟩∥)−1√) expected quantum walk steps and O(∥Dt|v⟩∥−1) expected reflections around |v⟩. This shows that quantum walks can accelerate the transient dynamics of Markov chains, complementing the line of results that proves the acceleration of their limit behavior. We show that this tool leads to speedups on random walk algorithms in a very natural way. Specifically we consider random walk algorithms for testing the graph expansion and clusterability, and show that we can quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, our quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, we use QFF for determining whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of s-t connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks
Near-Optimal Cayley Expanders for Abelian Groups
We give an efficient deterministic algorithm that outputs an expanding generating set for any finite abelian group. The size of the generating set is close to the randomized construction of Alon and Roichman [Alon and Roichman, 1994], improving upon various deterministic constructions in both the dependence on the dimension and the spectral gap. By obtaining optimal dependence on the dimension we resolve a conjecture of Azar, Motwani, and Naor [Azar et al., 1998] in the affirmative. Our technique is an extension of the bias amplification technique of Ta-Shma [Ta-Shma, 2017], who used random walks on expanders to obtain expanding generating sets over the additive group of ???. As a consequence, we obtain (i) randomness-efficient constructions of almost k-wise independent variables, (ii) a faster deterministic algorithm for the Remote Point Problem, (iii) randomness-efficient low-degree tests, and (iv) randomness-efficient verification of matrix multiplication
Gap Amplification for Reconfiguration Problems
In this paper, we demonstrate gap amplification for reconfiguration problems.
In particular, we prove an explicit factor of PSPACE-hardness of approximation
for three popular reconfiguration problems only assuming the Reconfiguration
Inapproximability Hypothesis (RIH) due to Ohsaka (STACS 2023). Our main result
is that under RIH, Maxmin Binary CSP Reconfiguration is PSPACE-hard to
approximate within a factor of . Moreover, the same result holds even
if the constraint graph is restricted to -expander for arbitrarily
small . The crux of its proof is an alteration of the gap
amplification technique due to Dinur (J. ACM, 2007), which amplifies the
vs. gap for arbitrarily small up to the vs.
gap. As an application of the main result, we demonstrate that
Minmax Set Cover Reconfiguration and Minmax Dominating Set Reconfiguratio} are
PSPACE-hard to approximate within a factor of under RIH. Our proof is
based on a gap-preserving reduction from Label Cover to Set Cover due to Lund
and Yannakakis (J. ACM, 1994). However, unlike Lund--Yannakakis' reduction, the
expander mixing lemma is essential to use. We highlight that all results hold
unconditionally as long as "PSPACE-hard" is replaced by "NP-hard," and are the
first explicit inapproximability results for reconfiguration problems without
resorting to the parallel repetition theorem. We finally complement the main
result by showing that it is NP-hard to approximate Maxmin Binary CSP
Reconfiguration within a factor better than .Comment: 41 pages, to appear in Proc. 35th Annu. ACM-SIAM Symp. Discrete
Algorithms (SODA), 202
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