VEST is W[2]-hard

In this short note, we show that the problem of VEST is $W[2]$-hard for parameter $k$. This strengthens a result of Matou\v{s}ek, who showed $W[1]$-hardness of that problem. The consequence of this result is that computing the $k$-th homotopy group of a $d$-dimensional space for $d > 3$ is $W[2]$-hard for parameter $k$.Comment: 5 pages, 2 figure

Balanced and almost balanced group presentations from algorithmic viewpoint

In this thesis we study algorithmic aspects of balanced group presentations which are finite presentations with the same number of generators and relations. The main motivation is that the decidability of some problems, such as the triviality problem, is open for balanced presentations. First, we summarize known results on decision problems for general finite presen- tations and we show two group properties which are undecidable even for balanced presentations - the property of "being a free group"' and the property of "having a finite presentation with 12 generators". We also show reductions of some graph problems to the triviality problem for group presentations, such as determining whether a graph is connected, k-connected or connected including an odd cycle. Then we show a reduction of the determining whether a graph with the same number of vertices and edges is a cycle to the triviality problem for balanced presentations. On the other hand, there is also a limitation of reduction to balanced presentations. We prove that there is no balanced presentation with two generators a, b|ap(m) bq(m) , ar(m) bs(m) for p(m), q(m), r(m), s(m) ∈ Z[m] which describes the trivial group if and only if m is odd. In the last part of this thesis, we describe a relation between group presentations and topology. In addition,..

Balancované a téměř balancované prezentace grup z algoritmického pohledu

In this thesis we study algorithmic aspects of balanced group presentations which are finite presentations with the same number of generators and relations. The main motivation is that the decidability of some problems, such as the triviality problem, is open for balanced presentations. First, we summarize known results on decision problems for general finite presen- tations and we show two group properties which are undecidable even for balanced presentations - the property of "being a free group"' and the property of "having a finite presentation with 12 generators". We also show reductions of some graph problems to the triviality problem for group presentations, such as determining whether a graph is connected, k-connected or connected including an odd cycle. Then we show a reduction of the determining whether a graph with the same number of vertices and edges is a cycle to the triviality problem for balanced presentations. On the other hand, there is also a limitation of reduction to balanced presentations. We prove that there is no balanced presentation with two generators a, b|ap(m) bq(m) , ar(m) bs(m) for p(m), q(m), r(m), s(m) ∈ Z[m] which describes the trivial group if and only if m is odd. In the last part of this thesis, we describe a relation between group presentations and topology. In addition,...V této práci se zabýváme algoritmickými vlastnostmi prezentací grup, což jsou konečné prezentace, kde počet generátorů a počet relací je stejný. Hlavní motivací je, že rozhodnutelnost některých problémů, např. zjistiť, zdali pre- zentace je prezentací triviální grupy (triviality problem), je pro balancované prezentace otevřená. Nejprve shrneme známé výsledky o rozhodovacích problémech pro ko- nečné prezentace a poté ukážeme dvě vlastnosti, které jsou nerozhodnutelné i pro balancované prezentace. Jedná se o vlastnosti "býti volnou grupou" a "mít prezentaci i na 12 generátorech". Dále ukážeme převody některých grafových problémů na triviality pro- blem. Např. rozhodování, zdali je graf souvislý, k-souvislý nebo souvislý ne- bipartitní. Také ukážeme převod rozhodování, zdali je graf se stejným po- čtem vrcholů a hran kružnice, na triviality problém pro balancované prezen- tace. Zamyslíme se také nad limity převodů na triviality problem pro balan- cované prezentace. Konkrétně ukážeme, že neexistuje balancovaná prezen- tace na dvou generátorech ⟨a, b|ap(m) bq(m) , ar(m) bs(m) ⟩, kde p(m), q(m), r(m), s(m) ∈ Z[m], která by popisovala triviální grupu právě tehdy, když m je liché. V poslední části této práce shrneme, jak prezentace grup souvisí s topo- logií. Doplněk k práci je také jednoduchý program, který...Katedra aplikované matematikyDepartment of Applied MathematicsFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Maximal point sets on discrete toric grid with no three colinear points

Let τ(Tm×n) denote maximal number of points on a discrete toric grid of the sizes m×n with no three colinear points. This thesis examines τ(Tm×n) for various m, n. It is a variant of the well-known no-three-in-line-problem. First, we present some previously known results. Then we generalize them in various directions. In particular we improve upper and lower bounds for cases which have not been solved in previous papers especially for cases when the sizes of the grid are prime powers. At the end we define the sequence (τ(Tm×n))n∈N and we prove that it is periodic for all fixed m.