Share Your Experience: Take the 2024 Developer Survey

# 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

76 questions
Filter by
Sorted by
Tagged with
728 views

• 9,419
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 ...
• 3,713
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 ...
• 1,166
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 ...
• 5,127
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 ...
• 664
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 $S$...
• 1,364
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 ...
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 ...
• 467
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, ...
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-...
• 901
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 ...
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 ...
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? ...
• 5,127
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. ...
• 9,458
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 ...
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 ...
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 ...
• 9,458
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 ...
• 91
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 ...
• 83
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 ...
• 2,153
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 ...
• 3,170
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 ...
172 views

• 9,458
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 ...
• 12.9k
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 ...
• 9,458
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 $H(s) = h(s_1)h(s_2)\dots h(s_n)$ ...
• 191
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 "...
• 630
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 ...
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 ...
• 1,453
139 views

• 12.9k