Counting Vanishing Matrix-Vector Products

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10494667" target="_blank" >RIV/00216208:11320/24:10494667 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1007/978-981-97-0566-5_24" target="_blank" >https://doi.org/10.1007/978-981-97-0566-5_24</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/978-981-97-0566-5_24" target="_blank" >10.1007/978-981-97-0566-5_24</a>

Result language

angličtina

Original language name

Counting Vanishing Matrix-Vector Products

Original language description

Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces: Let v is an element of Q(d) be a rational vector, (T-1, T-2... T-m) a list of d x d rational matrices, S is an element of Q(hxd) a rational matrix not necessarily square and k a parameter. The goal is to compute the number of ways one can choose k matrices T-i1, T-i2,..., T-ik from the list such that STik center dot center dot center dot T(i1)v = 0 is an element of Q(h). In this paper, we show that this problem is #W[2]-hard for parameter k. As a consequence, computing the k-th homotopy group of a d-dimensional 1-connected topological space for d &gt; 3 is #W[2]-hard for parameter k. We also discuss a decision version of the problem and its several modifications for which we show W[1]/W[2]-hardness. This is in contrast to the parameterized k-sum problem, which is only W[1]-hard (Abboud-Lewi-Williams, ESA&apos;14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized by the matrix dimensions and the order of the field.

Czech name

—

Czech description

—

Type

D - Article in proceedings

CEP classification

—

OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

Result was created during the realization of more than one project. More information in the Projects tab.

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2024

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Article name in the collection

WALCOM: ALGORITHMS AND COMPUTATION, WALCOM 2024

ISBN

978-981-9705-65-8

ISSN

0302-9743

e-ISSN

1611-3349

Number of pages

15

Pages from-to

335-349

Publisher name

SPRINGER-VERLAG SINGAPORE PTE LTD

Place of publication

SINGAPORE

Event location

Kanazawa

Event date

Mar 18, 2024

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

001207267500024