On the problem of approximating the eigenvalues of undirected graphs in probabilistic logspace

Dean Doron*, Amnon Ta-Shma

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

We introduce the problem of approximating the eigenvalues of a given stochastic/symmetric matrix in the context of classical spacebounded computation. The problem can be exactly solved in DET ⊆ NC2. Recently, it has been shown that the approximation problem can be solved by a quantum logspace algorithm. We show a BPL algorithm that approximates any eigenvalue with a constant accuracy. The result we obtain falls short of achieving the polynomially-small accuracy that the quantum algorithm achieves. Thus, at our current state of knowledge, we can achieve polynomially-small accuracy with quantum logspace algorithms, constant accuracy with probabilistic logspace algorithms, and no nontrivial result is known for deterministic logspace algorithms. The quantum algorithm also has the advantage of working over arbitrary, possibly non-stochastic Hermitian operators. Our work raises several challenges. First, a derandomization challenge, trying to achieve a deterministic algorithm approximating eigenvalues with some non-trivial accuracy. Second, a de-quantumization challenge, trying to decide whether the quantum logspace model is strictly stronger than the classical probabilistic one or not. It also casts the deterministic, probabilistic and quantum space-bounded models as problems in linear algebra with differences between symmetric, stochastic and arbitrary operators. We therefore believe the problem of approximating the eigenvalues of a graph is not only natural and important by itself, but also important for understanding the relative power of deterministic, probabilistic and quantum logspace computation.

Original languageEnglish
Title of host publicationAutomata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings
EditorsMagnus M. Halldorsson, Naoki Kobayashi, Bettina Speckmann, Kazuo Iwama
PublisherSpringer Verlag
Pages419-431
Number of pages13
ISBN (Print)9783662476710
DOIs
StatePublished - 2015
Event42nd International Colloquium on Automata, Languages and Programming, ICALP 2015 - Kyoto, Japan
Duration: 6 Jul 201510 Jul 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume9134
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference42nd International Colloquium on Automata, Languages and Programming, ICALP 2015
Country/TerritoryJapan
CityKyoto
Period6/07/1510/07/15

Funding

FundersFunder number
United States - Israel Binational Science Foundation2010120
Israel Science Foundation994/14

    Fingerprint

    Dive into the research topics of 'On the problem of approximating the eigenvalues of undirected graphs in probabilistic logspace'. Together they form a unique fingerprint.

    Cite this