Questions tagged [parameterized-complexity]

Computational complexity with respect to one or more parameters of the input (apart from its plain length as a string), which capture intrinsically difficult instances

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200 views

Relationship between complexity classes W[1] and NP?

I am trying to understand the connection between The W-hierarchy as presented in chapter 13 of this book by Cygan et al. and the notion of the NP problems. Is the existence of an FPT algorithm for a ...
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79 views

Proving that a problem is not FPT using reduction

In the Inclusive Vertex Cover problem, For a given graph $G=(V,E)$, each vertex $u\in V(G)$ has weight $u_{w} \in \mathbb{N}$ and value $u_{v}\in \mathbb{N}$. The value and weight of a set cover $S$ ...
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1answer
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What is T-star packing

What is a T-star packing and what is Proper Maximal 3-Star Packing? I read some definitions and can't understand "By a T-star we mean a complete bipartite graph K1,t for some t ≤ T. For an ...
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FPT algorithm for Knapsack

I tried to search whether Knapsack (the decision version) has a FPT algorithm, but didn't find anything on the topic. Can someone help with a reference?
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Petal-Deletion problem on O(k)

In Petal-Deletion, we are given an undirected graph G and k ∈ N, and the objective is to decide whether there exists a subset S ⊆ V (G) of size at most k such that G − S does not contain any 3-petal ...
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Approximate FPT and Approximate kernel for a decidable problem

I'm trying to prove the following: For every decidable parameterized problem Π, Π admits a fixed parameter tractable α-approximation algorithm if and only if Π has an α-approximate kernel. I'm trying ...
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57 views

Finding a kernel for 2-Bounded Degree Deletion problem of $O(k)$ vertices

In $2$ Bounded degree deletion problem, we're given an undirected graph $G$ and a positive integer $k$, and the task is to find at most $k$ such vertices whose removal decreases the maximum vertex ...
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64 views

Reduction rules to lower bound minimum degree of a graph

I'm trying to come up with a list of rules that return an equivalent instance to the following problem, while eliminating all vertices of degree 2 or less from the graph: Given a graph $G=(V,E)$, the ...
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Lite Feedback Vertex Set [duplicate]

In Lite Feedback Vertex Set, given an undirected graph G and k ∈ N, and the objective is to decide whether there exists a subset S ⊆ V (G) of size at most k such that every connected component in G − ...
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VS Problem algorithm

In CLOSED VERTEX COVER, we are given a graph G In CLOSED VERTEX COVER, we are given a graph 𝐺 where each vertex 𝑣∈𝑉(𝐺) has self-utility 𝑢𝑣∈ℕ and self-pollution 𝑝𝑣∈ℕ, and 𝑘,𝑈⋆,𝑃⋆∈ℕ. For each ...
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Algorithm for Variant of 0-1 Knapsack Problem

Variant of 0-1 Knapsack Problem is when you can choose exactly $k$ items from $n$ items, and $k$ is positive integer parameter that came in the input. Is there an algorithm with running time ...
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61 views

Feedback Vertex Set restricted to planar graphs

In Feedback Vertex Set, we are given an undirected graph $G$ and $k \in \mathbb{N}$, and the objective is to decide whether there exists a subset $S \subseteq V(G)$ of size at most $k$ such that $G-S$ ...
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Is there exist FPT algorithm for a variant of VC problem?

In CLOSED VERTEX COVER, we are given a graph $G$ where each vertex $v \in V(G)$ has self-utility $u_{v} \in \mathbb{N}$ and self-pollution $p_{v} \in \mathbb{N}$, and $k, U^{\star}, P^{\star} \in \...
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Parameterized Algorithms - vertex kernel for STAR DELETION

can someone gives some help here? some idea how to start? this is a question from a course called Parameterized Algorithms (not an homework. just a practise we got for the exam) (and here as a text ...
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Disjoint Lite Feedback Vertex Set

Disjoint Lite Feedback Vertex Set. given an undirected graph G and k ∈ N, and a Lite Feedback Vertex Set(LFVS) X of G of size ≤k+1 the objective is to decide whether there exists a subset S ⊆ V (G) of ...
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1answer
255 views

Dynamic Programming for Feedback Vertex Set - bounded treewidth

Saw it on another post that there is a way of solving FVS in polynomial time if the treewidth is constant, using dynamic programming?... If I'm given the treewidth of a graph, how do I solve it in ...
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FPT algorithm for a variant of Feedback Vertex Set

I am interested in a variant of the Feedback Vertex Set (FVS) problem. For an undirected graph $G$ and $k\in \mathbb{N}$ we need to decide if there is a subset $S \subseteq V(G)$ of size at most $k$ s....
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132 views

Claw-free graph - linear kernel

I'm having a hard time solving the problem below: In Claw-free problem, we are given a graph $G$ and $k$, and the objective is to decide whether there exists a subset $S \subseteq V (G)$ of size at ...
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$W$-hierarchy and parameterized search problems

I have two related questions: What are the ways to prove that a certain problem is in $W[t]$ in the W-hierarchy for parametrized complexity, except using the straight definition of boolean circuits? ...
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Constructing a crown graph given an independent set

A crown in a graph $G$ is a pair $(H, C)$, where $H \subseteq V(G)$ and $C \subseteq V(G)$ with $H ∩ C = ∅$ such that the following conditions hold: (a) The set of neighbors of vertices in $C$ is ...
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110 views

Color coding to get an FPT algoirthm for k disjoint triangles

The k-disjoint triangles problem is as follows: Input: A graph $G=(V,E)$ and an integer $k\in \mathbb{N}$ Output: Are there $k$ vertex-disjoint triangles in $G$? An FPT algorithm is presented here (...
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1answer
54 views

Time complexity for FPT algorithm

I'm studying the issue of FPT algorithms and came to the k-disjoint triangles problem as can be seen here on slide 60. The problem summary is given a graph G and variable k, are there k disjoint ...
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1answer
98 views

k disjoint triangles with graph splitting to two distinct groups

Please note that this question is different than this question. The $k$-disjoint triangles problem is as follows: Input: A graph $G=(V,E)$ and an integer $k\in \mathbb{N}$ Output: Are there $k$ ...
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Finding a kernel for 1-Bounded degree deletion of $O(k^2)$ vertices [duplicate]

In 1-BOUNDED DEGREE DELETION problem, we are given an undirected graph G and a positive integer $k$, and the task is to find at most $k$ such vertices whose removal decreases the maximum vertex degree ...
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1answer
104 views

Kernelization algorithm for the following problem

We are given an undirected graph $ G $ and a positive parameter $ k \geq 0 $. The problem is to decide if there exists a set $ S \subseteq V(G) $ of size at most $ k $ such that $ G − S $ does not ...
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1answer
258 views

FPT algorithm for 1-BDD

Given a graph $G = (V,E)$ and an integer $k$, the 1-BDD problem asks if there exists a subset $D$ of at most $k$ vertices such that the degree of any vertex in $G[V \setminus D]$ is at most one. Is ...
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183 views

Can someone tell a formal definition for this problem: k-disjoint triangle?

k-disjoint triangle "We consider the NP-complete problem of deciding whether an input graph on n vertices has k vertex-disjoint copies of a fixed graph H. " The above definition is the best ...
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28 views

Structural parametrization for weighted vertex cover

Let $G$ be a graph which is a tree with $\ell$ added edges. I wish to show that VWVC ((Vertex-)Weighted Vertex cover) is FPT with respect to $\ell$. In particular, I'd like an algorithm running in $O(...
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Tree Decomposition Construction with Balanced Separations: Why 2/3?

I am working through the book "Parameterized Algorithms" https://www.mimuw.edu.pl/~malcin/book/parameterized-algorithms.pdf and at the chapter about tree decomposition I am trying to ...
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1answer
66 views

Polynomial kernelization for Set Splitting

In a set system $(U, F)$, $F\subseteq \mathcal{P}(U))$, we say that a function $f: U \to \{0, 1\}$ is a coloring of $(U, F)$. A set in $F$ is split by $f$ if $F$ receives both colors. The Set ...
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Prerequisites for studying parametrized complexity

Which areas of CS/Math should one have mastered before diving into parametrized complexity?
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1answer
315 views

Finding a kernel for d-Bounded degree deletion

In $d$ Bounded degree deletion problem, we are given an undirected graph $G$ and a positive integer $k$, and the task is to find at most $k$ such vertices whose removal decreases the the maximum ...
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1answer
16 views

What does a kernel of size n,n^2 ,… mean?

So according to Wikipedia, In the Notation of [Flum and Grohe (2006)], a ''parameterized problem'' consists of a decision problem $L\subseteq\Sigma^*$ and a function $\kappa:\Sigma^*\to N$, the ...
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3answers
775 views

Is there an NP-hard problem for which no Fixed-Parameter Tractable algorithm exists?

Question Is there an NP-hard problem for which we can add a parameter1 to create a "natural"2 parametrised problem for which no FPT algorithm exists? The adding a parameter is needed ...
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1answer
375 views

Are there any known W[3] or W[3]-hard problems?

We are currently working on a variant of domination parameter and we have shown that it is in W[3] with regard to parameterized complexity. To show it is W[3]-complete, we must show the problem is W[3]...
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What about problems that are fixed parameter tractable with an algorithm that does not inspect the parameter?

A parameterized problem is a subset $L \subseteq \Sigma^* \times \mathbb N$, where $\Sigma$ is a finite alphabet. A parameterized problem is fixed parameter tractable, if it could be decided in time $...
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Sorting with high-latency compare

My basic setup is very simple: I'm trying to sort N items now, and later on I'll need to incrementally sort more items. The unique part of the problem is that my item comparison is not a computational ...
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1answer
266 views

Estimating P in Amdahl's Law theoretically and in practice

In parallel computing, Amdahl's law is mainly used to predict the theoretical maximum speedup for program processing using multiple processors. If we denote the speed up by S then Amdahl’s law is ...
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1answer
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Can the W hierarchy by defined by circuits having a satisfying assignment of weight at most k?

Traditionally, the $W$ hierarchy is defined around the problem of weighted circuit satisfiability. More precisely, the class $W[t]$ is defined as the closure under $\mathrm{fpt}$-reductions of the ...
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1answer
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Determine smallest possible parameter set for FPT

I am reading a book about complexity analysis and cannot find a way to solve a problem in that book. The problem is, that I do not understand how to determine the smallest possible parameters, given ...
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1answer
184 views

Parametrized reduction from 3-SAT to Independent Set to lower bound running time under ETH assumption

I want to prove that, assuming Exponential Time Hypothesis is true, there is no algorithm that solves Independent Set in $2^{o(|V|+|E|)}$ time. I want to apply the following strong parameterized many-...
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1answer
92 views

Is model checking PSpace-hard *in formula size*?

Sistla/Clarke proved [SC82] that the LTL model-checking problem is PSpace-complete. Sometimes people write that this problem is "PSpace-hard in $|\phi|$" (e.g. [LP85]). What does this mean formally? ...
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1answer
149 views

Why is dominating set in $W[2]$, but independent set in $W[1]$

In Parameterized Complexity the Independent Set Problem for a Parameter $k$ ist $W[1]$-complete, and Dominating set is $W[2]$-complete. Now the prototypical $W[1]$ problem is deciding by a single-tape ...
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1answer
179 views

Relationship between complexity classes XP and W[1]?

I am reading the introductory chapter in Parameterized Algorithms by Cygan et al. and I am having some problems with the distinction between complexity classes $\mathsf{W[1]}$ and $\mathsf{XP}$. ...
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1answer
182 views

Topics from theoretical computer science suitable for a bachelor (undergrate) thesis? [closed]

Is the field of theoretical computer science so complex, that it is just "too much" for a bachelor thesis? Unfortunately I haven't found any old thesises because the relevant chair of the university ...
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2answers
622 views

Exhaustive search algorithm solving vertex cover of size $k$ in time $2^{k}n^{O(1)}$?

In the wiki page of Vertex Cover, it is claimed that an exhaustive search algorithm can solve the problem(the decision version of vertex cover problem) in time $2^{k}n^{O(1)}$. Intuitively, with a ...
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1answer
96 views

Why does $W[1] = A[1]$ hold?

By definition, a parameterized problem $(Q, \kappa)$ is in $W[1]$ if it can be transformed into a combinatorial circuit $\varphi$ in polynomial time, such that the weft of $\varphi$ is 1. On the ...
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2answers
161 views

Minimum number of vertices whose removal makes the graph an independent set

It is known that finding an independent set (or a clique) of size at least $k$ in a graph is $W[1]$ hard, so it is unlikely that there is $f(k)\cdot n^{O(1)}$ time algorithm for finding an independent ...
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1answer
742 views

How is Vertex Cover reducable to Independent Set using parametrized reduction with parameter k?

We have the following Lemma and proof: Lemma 5.5. If $A$ if FPT, then $A\leq_{\mathrm{fpt}}$ Independent Set. Proof. We reduce $A$ to Independent Set parametrised by $k'$, where $k'$ is the size of a ...
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2answers
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What is a “slice” of a parameterized problem $(Q, \kappa)$?

In the book "Parameterized Complexity Theory" by J. Flum, and M. Grohe, there is a definition on page 7: Definition 1.10. Let $(Q, \kappa)$ be a parameterized problem and $\ell \in \mathbb{N}$. The $\...