Counting vanishing matrix-vector products

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10494671" target="_blank" >RIV/00216208:11320/24:10494671 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=AHc9TfA1Aa" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=AHc9TfA1Aa</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.tcs.2024.114877" target="_blank" >10.1016/j.tcs.2024.114877</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 & mldr;T-m) a list of dxd 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,& mldr;,T-ik from the list such that STik & ctdot;T(i1)v=0 is an element of Q(h).<br /> 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>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'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 the matrix dimensions and the order of the field.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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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

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