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Questions tagged [fixed-parameter-tractable]

algorithms for parameterized problems where the run-time is polynomial in the input size, but depends arbitrarily on the parameter

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FPT vs W[P] - Parameterized Complexity

In parametrized complexity, $\mathsf{FPT} \subseteq \mathsf{W}[1]$ $\subseteq \mathsf{W}[2]$ $\subseteq \ldots \subseteq \mathsf{W}[P]$. It is conjectured that each of the containments is proper. If $...
Uéverton dos santos souza's user avatar
19 votes
7 answers
2k views

Books/Lecture Notes on Parametrized Complexity

I would like to learn about Parametrized Complexity (both on the algorithmic side and on the hardness side). What books/lecture notes can I read on this subject?
Igor Shinkar's user avatar
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18 votes
4 answers
1k views

Open problems related to Graph isomorphism

Presently I am doing literature survey on Graph isomorphism (GI) problem. I would like to know some open questions related to the following What are the graph parameters for which fixed parameter ...
Kumar's user avatar
  • 2,044
18 votes
0 answers
421 views

Complexity of the homomorphism problem parameterized by treewidth

The homomorphism problem $\text{Hom}(\mathcal{G}, \mathcal{H})$ for two classes $\mathcal{G}$ and $\mathcal{H}$ of graphs is defined as follows: Input: a graph $G$ in $\mathcal{G}$, a graph $H$ in $...
a3nm's user avatar
  • 9,419
17 votes
1 answer
586 views

Clique-width expressions with logarithmic depth

When we are given a tree decomposition of a graph $G$ with width $w$, there are several ways in which we can make it "nice". In particular, it is known that it is possible to transform it into a tree ...
Michael Lampis's user avatar
17 votes
0 answers
602 views

Linear-time algorithm to test if clique number equals degeneracy bound?

Given a connected simple graph $G=(V,E)$, let $d$ denote its degeneracy and let $\omega$ denote the size of a maximum clique. A well-known bound on the clique number is $\omega\le d+1$, which is ...
Austin Buchanan's user avatar
15 votes
2 answers
1k views

Given a 4-cycle free graph $G$, can we determine if it has a 3-cycle in quadratic time?

The $k$-cycle problem is as follows: Instance: An undirected graph $G$ with $n$ vertices and up to $n \choose 2$ edges. Question: Does there exist a (proper) $k$-cycle in $G$? Background: For any ...
Michael Wehar's user avatar
15 votes
5 answers
1k views

Hardness of FPT problems

Vertex Cover can be easily reduced to Independent Set and vice versa. However, in the context of parameterized complexity, Independent set is harder than Vertex Cover. A kernel with $2k$ vertices ...
Nikhil's user avatar
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14 votes
5 answers
1k views

Exact Algorithms for r-Dominating Set on Bounded Treewidth Graphs

Given a graph, $G = (V, E)$, I want to find an optimal $r$-domination for $G$. That is, I want a subset $S$ of $V$ such that all vertices in $G$ are at a distance of at most $r$ from some vertex in $S$...
Nikhil's user avatar
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14 votes
3 answers
3k views

What is a natural problem in theory of computation?

In Stephen Cook's paper on the P vs NP problem,[1] he states the following [2]: Feasibility Thesis: A natural problem has a feasible algorithm iff it has a polynomial-time algorithm. My question ...
Curious Yogurt's user avatar
13 votes
2 answers
1k views

Relation between fixed parameter and approximation algorithm

Fixed parameter and approximation are totally different approaches to solve hard problems. They have different motivation. Approximation looks for faster result with approximate solution. Fixed ...
Prabu 's user avatar
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13 votes
1 answer
273 views

Elementary bounds on parameter in fixed-parameter tractability?

In the definition of (strong) fixed-parameter tractability, the time bound is an expression of the form $$f(k).p(|x|),$$ where the input instance is $(x,k)$ with parameter $k$, $p$ is a polynomial, ...
András Salamon's user avatar
12 votes
1 answer
307 views

Any results on binary boolean CSP beyond the fixed-parameter tractability of almost 2SAT problem?

Let $\varphi$ be a 2CNF formula and $k$ a nonnegative integer. It is proved in this paper that the problem of deciding whether one can delete at most $k$ clauses to make $\varphi$ satisfable, is fixed-...
Regularity's user avatar
11 votes
1 answer
441 views

Polynomial kernel for $k$-FLIP SAT on $3$-CNF formulas

The k-FLIP SAT parametrized problem is defined as: Input: a 3-CNF formula $\varphi$ with $n$ variables and a truth assignment $\sigma : [n] \to \{0,1\}$ Parameter: $k$ Question: can we transform the ...
Marzio De Biasi's user avatar
10 votes
1 answer
497 views

What is the motivation behind the definition of fixed parameter tractability?

Wikipedia writes: FPT contains the fixed parameter tractable problems, which are those that can be solved in time $f(k)\cdot|x|^{O(1)}$ for some computable function $f$. Typically, this ...
Douglas S. Stones's user avatar
10 votes
1 answer
954 views

Complexity of k-clique for hypergraphs

Classic Problem: Let a number $k$ be given. The $k$-clique problem is as follows. Given a graph $G$, does there exist a subset $S$ of $k$ vertices so that any two vertices of $S$ are adjacent? ...
Michael Wehar's user avatar
10 votes
2 answers
452 views

Which graph problems are $W[1]$-Hard on directed(/weighted) graphs but FPT on undirected(/unweighted) graphs?

Following the equivalent questions regarding NP-Completeness (see the weight question and the directed question), I was wondering how parameterized problems are affected by these attributes. ...
R B's user avatar
  • 9,458
10 votes
1 answer
633 views

Conjecture: All FPT NP-complete languages are fixed-parameter-isomorphic

Berman–Hartmanis conjecture: all NP-complete languages look alike, in the sense that they can be related to each other by polynomial time isomorphisms[1]. I am interested in a more fine-grained ...
user avatar
10 votes
0 answers
275 views

Have people looked for parameterized algorithms for problems that are not in NP?

Are there problems that are not in NP (e.g., NEXP-complete problems) but admit FPT algorithms for a reasonable parameterization (and specifically, the standard parameterization of a problem -- the ...
PHD candidate's user avatar
9 votes
5 answers
1k views

W[1]-hard problems with FPT time approximation algorithms

I'm looking for problems that are hard to solve in FPT time but has an approximation algorithm. That is, problems that are: R1. W[1]-hard. R2. Admit a (preferably constant) approximation algorithm ...
R B's user avatar
  • 9,458
9 votes
1 answer
223 views

What is the best upper bound on the running time of the graph minor algorithm?

A cornerstone of the graph minor theory is an algorithm that, given undirected graphs $G, H$, runs in time $f(|H|)poly(|G|)$, and determines whether $H$ is a minor of $G$ or not. It has been obtained ...
Michal's user avatar
  • 91
8 votes
3 answers
389 views

Is parametrized maximum independent clauses problem in FPT?

Parametrized maximum independent clauses problem: Input : A r-CNFSAT formula F having n variables and m clauses, k Ques : Does there exists at least k clauses such that they are mutually independent ...
Gaurav's user avatar
  • 83
8 votes
2 answers
263 views

Nonstandard dual parametrization of graph problems

One fundamental result in parameterized complexity of graph problems is that VERTEX COVER parameterized by the solution size $k$ is fixed-parameter-tractable (FPT). On the other hand, when ...
Florent Foucaud's user avatar
8 votes
2 answers
502 views

Implications of a problem being in XP when parameterized by diameter

Let $X$ be an NP-complete graph problem. Suppose $X$ is solvable in polynomial time on graphs of bounded diameter. In other words, $X$ parameterized by diameter is in XP. (Recall a problem is in XP if ...
Juho's user avatar
  • 3,170
8 votes
1 answer
589 views

FPT algorithm for mixed integer program

It is known that every integer linear program parameterized by the number of variables is FPT (fixed parameter tractable). Is every mixed integer program parameterized by the number of integer ...
Piotr Skowron's user avatar
8 votes
0 answers
172 views

Is 4-colors precoloring extension for planar graphs fixed parameter tractable?

Given a planar graph $G=(V,E)$, there exists a quadratic algorithms for 4-coloring $G$ (and $G$ is surely 4-colorable). Assume you are given a set of $k$ constraints of the form "$v_i \text{ is ...
R B's user avatar
  • 9,458
7 votes
3 answers
538 views

Algebraic formulation for packing problem

My question is regarding the algebraic formulation for packing problems in graphs. Taking an example, suppose I am interested in the problem of finding if there is a packing of k edge disjoint ...
Nikhil's user avatar
  • 664
7 votes
3 answers
738 views

Hard problems for bounded vertex cover

We know that list coloring problem is W[1]-hard when parameterized by vertex cover. Are there any other problems which are also W[1]-hard parameterized by vertex cover?
Kumar's user avatar
  • 2,044
7 votes
2 answers
791 views

Variants of Cluster-Vertex-Deletion problem

The Unweighted Cluster-Vertex-Deletion problem is the following: Input: An undirected graph G = (V, E) and a nonnegative number k Output: Is there a subset X ⊆ V with |X| ≤ k such that deleting all ...
user2582's user avatar
  • 531
7 votes
2 answers
225 views

Is it known whether counting $q$-dimensional $p$-matching is $\#W[1]$-Hard?

The $q$-Dimensional $p$-Matching is defined as follows: Given disjoint universes $U_1,\ldots,U_q$, think of an element in $U_1\times\ldots\times U_q$ as a set that contains exactly one element from ...
R B's user avatar
  • 9,458
7 votes
1 answer
171 views

Reducing Parameterized Problems (whose solution size can be "large") to W[i]-complete problems (for fixed i)

Note: Originally, this question was asked via a comment in this question, but was asked to post a separate question. :) I'm looking for any known reductions of the following: Given a parameterized ...
Haden Hooyeon Lee's user avatar
6 votes
2 answers
267 views

Parametrized complexity of the 2-Long Paths Problem

Consider the following problem: Let $G=(V,E)$ be a graph, $s,t\in V$ vertices and $k\in\mathbb N$ an integer parameter. The 2-Long Paths Problems asks whether there exist two disjoint paths from $s$...
R B's user avatar
  • 9,458
6 votes
1 answer
246 views

Easy decision hard counting Parametrized

It is known that counting perfect matchings in a bipartite graph is #P-complete. On the other hand, finding a perfect matching belongs in P. Is there a problem, that exhibits the same behavior in ...
Paramar's user avatar
  • 447
5 votes
2 answers
422 views

On integer programming

Integer programming is NP-hard. What is the status of integer programming problem that decides between existence of $\leq1$ solution and $>1$ solutions (note $0$ solutions falls in $\leq1$ ...
Turbo's user avatar
  • 12.9k
5 votes
1 answer
563 views

Complexity of SAT parameterized by treewidth

Many papers state that Boolean satisfiability is in FPT when parameterized by primal, dual, or incidence treewidth. What are the best known time complexities of these parameterized algorithms? In ...
Laakeri's user avatar
  • 1,786
5 votes
0 answers
132 views

Is minimum weight simple cycles through specified vertics fixed parameter tractable?

The problem formulation is as follows: Input: Undirected graph $G=(V,E)$, a set of vertices $S\subseteq |V|$, a weight function $w:E\to \mathbb{R}$ and a threshold $T\in \mathbb{R}$. Parameter: $|S|=...
R B's user avatar
  • 9,458
4 votes
1 answer
631 views

On Random Self-reducible properties

Permanent is random self-reducible. $\mathsf{SAT}$ is not random self-reducible since otherwise the polynomial hierarchy collapses to $\mathsf{\Sigma_3}$. 1) Is $k$-sum random self-reducible? That ...
Turbo's user avatar
  • 12.9k
4 votes
1 answer
434 views

Vertex disjoint simple paths of length k

A lot of effort has been invested in finding simple k-paths, as well as in finding vertex disjoint paths. Is there any known parametrized algorithm that given a graph $G=(V,E)$, decides whether there ...
R B's user avatar
  • 9,458
4 votes
0 answers
95 views

Fixed-parameter tractability of string homomorphism

String homomorphism is a function $h: \Sigma \to \Sigma^*$, which naturally defines a homomorphism on strings from $\Sigma^*$ with respect to concatenation. We denote $H(s) = h(s_1)h(s_2)\dots h(s_n)$ ...
Ivan Smirnov's user avatar
3 votes
3 answers
1k views

Best parameterized algorithm for maximum clique

I have seen the basic algorithm for the maximum clique problem parameterized by the maximum degree at an algorithms course. However, I struggle to find anything better. Searching for things like "...
user2316602's user avatar
3 votes
1 answer
643 views

What is the best known FPT result for 3-hitting set?

My research problem involves solving a special instance of the 3-Hitting Set problem, and I was wondering whether my result is actually significant (i.e. if it is better than the best known result for ...
user1441057's user avatar
3 votes
1 answer
276 views

Complement problems are not in the same class in parametrized complexity hierarchy? If not in $P$

By "complement problems", I mean the two problems' objective functions are complement. For example, the vertex cover and its complement independent set in this sense. For a graph $G(V,E)$, their ...
Peng Zhang's user avatar
  • 1,453
3 votes
1 answer
139 views

Consequences of faster parameterized integer programming

Integer programming in $k$ variables can be done in $k^{O(k)}$ time and $O(k^c)$ space. Is there any consequence if it can be done in $k^{O(k^\alpha)}$ time and $O(k^c)$ space for some $\alpha\in(0,1)...
Turbo's user avatar
  • 12.9k
3 votes
1 answer
212 views

Solving Feedback Vertex Set (FVS) in FPT time $5^k$ with iterative compression?

I understand that Disjoint Feedback Vertex Set (= looking for a solution $X$ of size $k$ given a solution $W$ of size $k+1$ s.t. $X \subseteq V \setminus W$ ) can be solved in time $4^k poly(n)$, see ...
yassin's user avatar
  • 161
3 votes
1 answer
163 views

Parameterized algorithm when the parameter is not known in advance?

In the setting of parameterized algorithms, we are typically given the problem instance as well as the value of the parameter. However, it seems like in applications the value of the parameter should ...
Karagounis Z's user avatar
3 votes
1 answer
149 views

Is it possible to approximate Maximum Independent Set in $O(2^k\text{poly}(n))$ time?

We know that MIS is hard to approximate within a $n^{1-\epsilon}$ factor in polynomial time and that it is $W[1]$-hard and thus unlikely to admit a $f(k)\text{poly}(n)$ time exact algorithm. (here, $k$...
JFK's user avatar
  • 31
3 votes
0 answers
89 views

Exact FPT Algorithm for Continuous Euclidean $k$-Means

The continuous Euclidean $k$-means problem is defined as follows: Given a set $X$ of $n$ points in $d$ dimensional Euclidean space $\mathbb{R}^{d}$. Given a parameter $k>0$, find a partitioning $P$ ...
Inuyasha Yagami's user avatar
3 votes
0 answers
151 views

What are the consequences if $W[i]=W[i-1]$?

$FPT=W[1]$ does not collapse the $W$ hierarchy however falsifies $ETH$ belief. Is there non-trivial consequence if $W[i]=W[i-1]$ and any other consequence at $W[1]$?
Turbo's user avatar
  • 12.9k
3 votes
0 answers
105 views

Fixed parameter Integer Programming circuit depth complexity

It is well known Lenstra's and Kannan's algorithm achieves $n$-variable parameter $L$-bit integer programming solvability in $O(n^nL)$ time and $O(L)$ space. If implemented as an arithmetic circuit ...
Turbo's user avatar
  • 12.9k
3 votes
0 answers
136 views

Multiple knapsack fpt?

It was an open question whether multiple knapsack is fpt wrt standard parameter. Since at SODA 2009 Jansen has presented an EPTAS for multiple knapsack and an EPTAS implies the existense of an fpt ...
Thomas's user avatar
  • 57