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Parameterized Intractability of Even Set and Shortest Vector Problem

Authors:

Arnab Bhattacharyya,

Édouard Bonnet,

Suprovat Ghoshal,

Pasin Manurangsi,

Dániel MarxAuthors Info & Claims

Journal of the ACM (JACM), Volume 68, Issue 3

Article No.: 16, Pages 1 - 40

https://doi.org/10.1145/3444942

Published: 22 March 2021 Publication History

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Abstract

The -Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over

, which can be stated as follows: given a generator matrix

and an integer

, determine whether the code generated by

has distance at most

, or, in other words, whether there is a nonzero vector

such that

has at most

nonzero coordinates. The question of whether

-Even Set is fixed parameter tractable (FPT) parameterized by the distance

has been repeatedly raised in the literature; in fact, it is one of the few remaining open questions from the seminal book of Downey and Fellows [1999]. In this work, we show that

-Even Set is W[1]-hard under randomized reductions.

We also consider the parameterized -Shortest Vector Problem (SVP), in which we are given a lattice whose basis vectors are integral and an integer

, and the goal is to determine whether the norm of the shortest vector (in the

norm for some fixed

) is at most

. Similar to

-Even Set, understanding the complexity of this problem is also a long-standing open question in the field of Parameterized Complexity. We show that, for any

,

-SVP is W[1]-hard to approximate (under randomized reductions) to some constant factor.

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Bandyapadhyay SFomin FGolovach PSimonov K(2023)Parameterized Complexity of Feature Selection for Categorical Data ClusteringACM Transactions on Computation Theory10.1145/360479715:3-4(1-24)Online publication date: 12-Dec-2023
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Bukh BKarthik CNarayanan B(2022)Applications of Random Algebraic Constructions to Hardness of Approximation2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00032(237-244)Online publication date: Feb-2022
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Index Terms

Parameterized Intractability of Even Set and Shortest Vector Problem
1. Theory of computation

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 68, Issue 3

June 2021

244 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/3456663

Editor:
Éva Tardos
Cornell University, United States

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Copyright © 2021 Association for Computing Machinery.

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Publication History

Published: 22 March 2021

Accepted: 01 December 2020

Revised: 01 December 2020

Received: 01 September 2019

Published in JACM Volume 68, Issue 3

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Funding Sources

Ramanujan Fellowship DSTO
Indo-US Joint Center for Pseudorandomness in Computer Science while at Indian Institute of Science
European Research Council (ERC)
ERC-CoG
JSPS KAKENHI
JST ERATO

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Cited By

Bandyapadhyay SFomin FGolovach PSimonov K(2023)Parameterized Complexity of Feature Selection for Categorical Data ClusteringACM Transactions on Computation Theory10.1145/360479715:3-4(1-24)Online publication date: 12-Dec-2023
https://dl.acm.org/doi/10.1145/3604797
Bennett H(2023)The Complexity of the Shortest Vector ProblemACM SIGACT News10.1145/3586165.358617254:1(37-61)Online publication date: 1-Mar-2023
https://dl.acm.org/doi/10.1145/3586165.3586172
Bukh BKarthik CNarayanan B(2022)Applications of Random Algebraic Constructions to Hardness of Approximation2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00032(237-244)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00032
Arvind VKöbler JKuhnert STorán J(2021)Parameterized Complexity of Small Weight Automorphisms and IsomorphismsAlgorithmica10.1007/s00453-021-00867-y83:12(3567-3601)Online publication date: 1-Dec-2021
https://dl.acm.org/doi/10.1007/s00453-021-00867-y

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