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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But why *is* $FPT$ a subset of $W[1]$?
$FPT$ is the set of parameterized problems that are fixed-parameter tractable. If $L_{w,h}$ is the language associated to boolean circuits of weft $w$ and depth $h$, then $W[t]$ is the set of ...
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vertex-deletion and edge-deletion parameters
Let $\mathcal{G}$ be a graph class. For any class $\mathcal{H}$, we know that the minimal number of vertices that has to be removed from $G\in \mathcal{H}$ such that we get a graph from $\mathcal{G}$ ...
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Is there such a thing as $coW[1]$-hardness?
I have a problem $\mathsf{A}$ and I would like to analyze its (parameterized) computational complexity.
I found a parameterized reduction from the complement of the independent set ($\mathsf{coIS}$) ...
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No parameterised reduction for a problem indicates FPT or not?
I am currently working on parameterized complexity, especially on the hard proofs. There is a problem that I am currently working on, denoted by $P$ and a parameter $x$, I discovered that there is no ...
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Can a PTAS be called one if it is parameterized by one of the problem inputs (in addition to ε)?
I.e. is it right to say "a PTAS parameterized by sth"?
Is it unusual, and is it correct?
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Constant value NP-complete vs W[1]-hard
I am a research scholar currently working in parameterized algorithms. I am studying the complexity of a problem (say $P$) for $\Delta_{10}$ graphs and was able to provide a reduction from a known NP-...
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Computing distance to clique in FPT time
I am a research scholar, and I currently work in parameterized algorithms. My current work involves proving that a problem is FPT for the parameter distance to clique. Although it is known that ...
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Query on kernel existence between related parameters
I am a research scholar working on parameterized complexity. For more information on parameterized complexity please refer to this. I am exploring on the tractability of an NP-complete problem $P$ for ...
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Fpt algorithm for hypergraph basis problem
While working through the book Parameterized Complexity Theory of Flum and Grohe, I encountered exercise 1.42:
Let $H = (V, E)$ be a hypergraph. A basis of $H$ is a set $S$ of subsets of $V$ with the ...
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Query on distance to parameter in parameterized complexity
I am a research scholar working in parameterized algorithms. For more information on parameterized complexity please refer to this. Recently, I have come across a problem which is known to be NP-...
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Tractability w.r.t. multiple parameters
I am working on the decision version of an NP-complete problem. The problem is known to be fixed parameter tractable(FPT) with respect to the solution size $k$ as the parameter.
If I consider another ...
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Relationship between the parameters tree width and the maximum degree
I am working on parameterized complexity and started exploring on various structural parameters. The problem I am working on is known to be W[1]-hard parameterized by treewidth of the input graph and ...
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CNF – satisfy at most a fixed number of clauses
I'm working on this task:
Prove that the following problem can be solved in time $2^{k} \cdot \Vert \varphi \Vert^{\mathcal{O}(1)}$: given a boolean formula $\Vert \varphi \Vert$ in CNF, decide ...
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Why do we care about computable functions in parametrized complexity definitions?
In a classical definition of the FPT complexity class, for a parametrized problem $L\subseteq \Sigma^* \times \mathbb{N}$ we require an algorithm, solving an instance $(G,k)$ in time $f(k)\cdot n^{O(1)...
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Subgraph Isomorphism with Same Number of Nodes
I am looking at a specific variant of subgraph isomorphism:
Instance A graph $G = (V_G, E_G)$ and a target graph $H = (V_H, E_H)$ such that $|V_G| = |V_H|$.
Question Is there a subgraph $G' = (V'_G, ...
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Bounded search tree: k-Vertex cover with $\Delta(G) = 2$
I am studying the book "Parametrized Algorithms" and it suggests a bounded search tree algorithm for k-Vertex Cover. Basically we look at a vertex v and say :
Either v will cover it's ...
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How to find parameter sets for a big-O expression to be fixed-parameter tractable?
I've been stuck on the following assignment taken from Cognition and Intractability: A Guide to Classical and Parameterized Complexity Analysis:
Imagine that the following big-O expressions ...
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Does FPT allow for doubling the parameter?
I have recently come across a result that showed that a given problem is in FPT when parameterized by the treewidth of a graph. However, they did this by showing that the problem is in FPT when ...
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First-order model checking is not fixed parameter tractable on general graphs
I read that the problem of first-order model checking is believed to be not fixed parameter tractable on general graphs.
Why is this the case? Would be happy about some reference
Thanks in advance!
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First-order model checking on general graphs is intractable
I read that the first-order model checking problem is intractable on general graphs.
How is this shown? Would be happy about some reference!
Thanks in advance
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most cost-effective route w.r.t. gas in a labelled graph
Consider a car that can hold gas to travel a distance of $c \in N$ kilometers (its capacity) on a full tank that's initially empty. The car starts in node $s \in V$ of a graph. Each vertex $V_i$ of ...
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Runtime analysis with multiple parameters: case study with heaps in Huffman coding
While studying a book on algorithms, I came across a question that asked about essentially $d$-ary Huffman coding, where the codeword alphabet has $d$ symbols (the usual case has $d=2$, with symbols $...
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The length of the formula in Monadic second-order logic
The Courcelle's theorem as following:
Obviously, we need the length of the $\varphi$~(that is $||\varphi||$)~in Theorem. However, how to caculate the length of $\varphi$?
For example, $$\begin{...
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Cost of finding optimal elimination order in a planar tensor network?
Suppose we are computing a sum over $n$ factors which can be represented as a planar tensor network. What is the complexity of finding an optimal elimination order?
For example, take the following ...
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The relation of W[1]-hard and Para-NP-hard
Is it possible that a problem is both W[1]-hard and Para-NP-hard?
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Complexity difference of Vertex cover and Vertex coloring regarding parametrized algorithms
I stumbled upon some problem in my understanding of the complexity classes FPT and XP.
According to Wikipedia (and the Book "Parameterized Algorithms") we know the following about the Vertex ...
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Proving FPT is strictly contained in XP
In their book Fundamentals of Parameterized Complexity, Downey and Fellows claim (in chapter 27.1) that $\mathrm{FPT}\subsetneq \mathrm{XP}$, and that this is a "basic result" that follows ...
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Determining if a maximum Independent set is in FPT when parameterized
I'm confused to if there exists an algorithm that can solve the below maximum independent set in polynomial time where it is parameterized by a k value.
maximum independent set
Input: A graph G with ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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