TY - JOUR

T1 - Pushing the Online Boolean Matrix-vector Multiplication conjecture off-line and identifying its easy cases

AU - Gąsieniec, Leszek

AU - Jansson, Jesper

AU - Levcopoulos, Christos

AU - Lingas, Andrzej

AU - Persson, Mia

N1 - Funding Information:
Thanks go to the anonymous reviewers for their valuable comments. The authors were supported in part by Swedish Research Council grant 621-2017-03750 . JJ was also supported by PolyU Fund 1-ZE8L .
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/6

Y1 - 2021/6

N2 - Henzinger et al. posed the so-called Online Boolean Matrix-vector Multiplication (OMv) conjecture and showed that it implies tight hardness results for several basic dynamic or partially dynamic problems [STOC'15]. We first show that the OMv conjecture is implied by a simple off-line conjecture that we call the MvP conjecture. We then show that if the definition of the OMv conjecture is generalized to allow individual (i.e., it might be different for different matrices) polynomial-time preprocessing of the input matrix, then we obtain another conjecture (called the OMvP conjecture) that is in fact equivalent to our MvP conjecture. On the other hand, we demonstrate that the OMv conjecture does not hold in restricted cases where the rows of the matrix or the input vectors are clustered, and develop new efficient randomized algorithms for such cases. Finally, we present applications of our algorithms to answering graph queries.

AB - Henzinger et al. posed the so-called Online Boolean Matrix-vector Multiplication (OMv) conjecture and showed that it implies tight hardness results for several basic dynamic or partially dynamic problems [STOC'15]. We first show that the OMv conjecture is implied by a simple off-line conjecture that we call the MvP conjecture. We then show that if the definition of the OMv conjecture is generalized to allow individual (i.e., it might be different for different matrices) polynomial-time preprocessing of the input matrix, then we obtain another conjecture (called the OMvP conjecture) that is in fact equivalent to our MvP conjecture. On the other hand, we demonstrate that the OMv conjecture does not hold in restricted cases where the rows of the matrix or the input vectors are clustered, and develop new efficient randomized algorithms for such cases. Finally, we present applications of our algorithms to answering graph queries.

KW - Boolean matrix

KW - Dynamic graph problems

KW - Online computation

KW - Product of matrix and vector

KW - Time complexity

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U2 - 10.1016/j.jcss.2020.12.004

DO - 10.1016/j.jcss.2020.12.004

M3 - Article

AN - SCOPUS:85099348676

SN - 0022-0000

VL - 118

SP - 108

EP - 118

JO - Journal of Computer and System Sciences

JF - Journal of Computer and System Sciences

ER -