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
Questions tagged [parameterized-complexity]
114 questions
13
votes
1 answer
Why are all problems in FPTAS also in FPT?
According to the Wikipedia article on polynomial-time approximation schemes:
All problems in FPTAS are fixed-parameter tractable.
This result surprises me - these classes seem to be totally different from one another. FPTAS characterizes problems…
templatetypedef
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9
votes
2 answers
Courcelle's Theorem: Looking for papers
I am looking for an easy and introductory paper on the proof of Courcelle's Theorem. I am also interested in its connection to parameterized complexity regarding the treewidth.
I am only a beginner in this field.
Any suggestions?
Laura
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9
votes
1 answer
Find which vertices to delete from graph to get smallest largest component
Given a graph $G = (V, E)$, find $k$ vertices $\{v^*_1,\dots,v^*_k\}$, which removal would result in a graph with smallest largest component.
I assume for large $n = |V|$ and large $k$ the problem is difficult (NP-hard), but I am interested in…
MindaugasK
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8
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1 answer
Equivalence between two definitions of Tree width
Treewidth :
1) By chordal graphs : size of the largest clique $(\omega (G))$ - 1 in a chordal completion of the graph $G$.
2) By tree decomposition :
A tree decomposition of $G = (V , E)$ consists of a tree $T$ (on a different node set from $G$),…
user35837
7
votes
1 answer
Kernels in parameterized complexity
Can anyone explain me what (problem-)kernels are and what's the use of them? My slides say:
The kernel of a parameterized problem $L$ is a transformation $(x,k) \mapsto (x',k')$ such that:
$(x,k) \in L \Leftrightarrow (x',k') \in L$
$|x'| \leq…
Peter W
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7
votes
1 answer
Does $\#W$[1]-hardness imply approximation hardness?
Let $\Pi$ be a parametrized counting problem, where the parameter is the solution cost, e.g. counting the number of $k$-sized vertex cover in a graph, parametrized by $k$.
Assume that $\Pi$ is $\#W$[1]-complete (a known problem for example would be…
R B
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7
votes
1 answer
NP complete problems that are solvable in polynomial time if the input (e.g. number of variables) is fixed?
I have seen some problems that are NP-hard but polynomially solvable in fixed dimension.
Examples, I think, are Knapsack that is polynomial time solvable if the number of items is fixed and Integer Linear Programming with fixed number of variables…
user2145167
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6
votes
1 answer
Implementing general vertex folding procedure in an undirected graph
I'm implementing the algorithm presented in "Improved Parameterized Upper Bounds for Vertex Cover" paper (PDF).
I'm a bit stumped by the General-Fold procedure.
What it should do is reduce the number of vertices (and) edges in the graph by finding…
helluin
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6
votes
0 answers
Reduction from clique to bag automata
I am trying to figure out a reduction to prove $W[1]$-hardness for this, but I am having significant trouble. Here is the problem:
Bag Automaton:
A non deterministic finite state automaton $M=(Q,I,s,F,d)$. $Q$ is the set of states, $I$ is the set…
YugiohMishima
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6
votes
0 answers
Decomposition of graphs that uses centers
Do you know of any kind of decomposition of graphs that involves centers, especially in the context of parametrized complexity? If so, please provide some reference. If not, do you see any reason (other than the potentially large size of centers)…
frafl
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6
votes
1 answer
$1+\epsilon$ approximation for inapproximable problems
I am currently confused by the following situation:
1) The metric $k$-center problem is inapproximable in polynomial time within $2-\epsilon$ unless $P=NP$.
2) The metric $k$-center problem can approximated within $1+\epsilon$ in time $O(k^{O(k/…
jack
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5
votes
2 answers
What does the complexity class $\mathsf{XP}$ stand for?
$\mathsf{XP}$ is the class of problems with input length $n$ and parameter $k$ than can be solved in $O(n^{f(k)})$ time, where $f$ is a computable function. It's described on the complexity zoo page as "Fixed-parameter Tractable for Each Parameter",…
Will Bolden
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5
votes
1 answer
FPT algorithm for point line cover
In the "Covering Things with Things" paper from Langerman and Morin, they mention the BST-Dim-Set-Cover, which is a FPT algorithm for point-line-cover, at page 666.
The algorithm chooses each point p from a set S and then subdivides it into…
testTester
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5
votes
0 answers
Sokoban with only $k$ boxes
Note: I have posted a hugely expanded version of this question on cstheory.
Since a Sokoban instance with only $k$ boxes has at most $n^{O(k)}$ possible states, the problem lies in $\operatorname{NSPACE}$$[O(k \cdot \log (n))]$. Since the region
X …
user12859
5
votes
1 answer
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-one reduction $f$ from 3-Sat to Independent set.…
MLStudent
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