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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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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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2 votes
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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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Parameterized complexity of 3-SAT

Is the 3-SAT problem with $s$ variables and $t$ clauses FPT, parameterized by $s+t$, or W[1]-hard, or para-NP-hard?
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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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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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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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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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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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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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4 votes
1 answer
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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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2 votes
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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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1 vote
1 answer
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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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2 answers
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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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4 votes
1 answer
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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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0 votes
1 answer
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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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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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4 votes
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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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2 votes
1 answer
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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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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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3 votes
1 answer
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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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5 votes
1 answer
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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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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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