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Electronic Colloquium on Computational Complexity

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Reports tagged with Parameterized Complexity:
TR97-001 | 8th January 1997
Marco Cesati, Luca Trevisan

On the Efficiency of Polynomial Time Approximation Schemes

A polynomial time approximation scheme (PTAS) for an optimization
problem $A$ is an algorithm that on input an instance of $A$ and
$\epsilon > 0$ finds a $(1+\epsilon)$-approximate solution in time
that is polynomial for each fixed $\epsilon$. Typical running times
are $n^{O(1/\epsilon)}$ or $2^{1/\epsilon^{O(1)}} ... more >>>

TR97-006 | 31st January 1997
Marco Cesati, Miriam Di Ianni

Parameterized Parallel Complexity

Comments: 1

We consider the framework of Parameterized Complexity, and we
investigate the issue of which problems do admit efficient fixed
parameter parallel algorithms by introducing two different
degrees of efficiently parallelizable parameterized problems, PNC
and FPP. We sketch both some FPP-algorithms solving natural
parameterized problems and ... more >>>

TR00-080 | 24th July 2000
Marco Cesati

Perfect Code is W[1]-complete

We show that the parameterized problem Perfect Code belongs to W[1].
This result closes an old open question, because it was often
conjectured that Perfect Code could be a natural problem having
complexity degree intermediate between W[1] and W[2]. This result
also shows W[1]-membership of the parameterized problem Weighted
more >>>

TR01-023 | 13th March 2001
Jochen Alber, Henning Fernau, Rolf Niedermeier

Parameterized Complexity: Exponential Speed-Up for Planar Graph Problems

Revisions: 1

A parameterized problem is called fixed parameter tractable
if it admits a solving algorithm whose running time on
input instance $(I,k)$ is $f(k) \cdot |I|^\alpha$, where $f$
is an arbitrary function depending only on~$k$. Typically,
$f$ is some exponential function, e.g., $f(k)=c^k$ for ... more >>>

TR04-027 | 20th February 2004
Henning Fernau

Parametric Duality: Kernel Sizes and Algorithmics

We derive the first lower bound results on kernel sizes of parameterized problems. The same idea also allows us to sometimes "de-parameterize"
parameterized algorithms.

more >>>

TR06-011 | 2nd January 2006
Yijia Chen, Martin Grohe

An Isomorphism between Subexponential and Parameterized Complexity Theory

We establish a close connection between (sub)exponential time complexity and parameterized complexity by proving that the so-called miniaturization mapping is a reduction preserving isomorphism between the two theories.

more >>>

TR06-072 | 25th February 2006
Henning Fernau

Parameterized Algorithms for Hitting Set: the Weighted Case

We are going to analyze simple search tree algorithms
for Weighted d-Hitting Set. Although the algorithms are simple, their analysis is technically rather involved. However, this approach allows us to even improve on elsewhere published algorithm running time estimates for the more restricted case of (unweighted) d-Hitting Set.

... more >>>

TR07-001 | 19th November 2006
Stefan S. Dantchev, Barnaby Martin, Stefan Szeider

Parameterized Proof Complexity: a Complexity Gap for Parameterized Tree-like Resolution

Revisions: 2

We propose a proof-theoretic approach for gaining evidence that certain parameterized problems are not fixed-parameter tractable. We consider proofs that witness that a given propositional formula cannot be satisfied by a truth assignment that sets at most k variables to true, considering k as the parameter. One could separate the ... more >>>

TR07-091 | 10th September 2007
Martin Grohe

Logic, Graphs, and Algorithms

Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle's Theorem, which states that all graph properties definable in monadic second-order logic ... more >>>

TR07-106 | 10th September 2007
Yijia Chen, Martin Grohe, Magdalena Grüber

On Parameterized Approximability

Combining classical approximability questions with parameterized complexity, we introduce a theory of parameterized approximability.
The main intention of this theory is to deal with the efficient approximation of small cost solutions for optimisation problems.

more >>>

TR07-137 | 6th November 2007
Yijia Chen, Jörg Flum, Moritz Müller

Lower Bounds for Kernelizations

Among others, refining the methods of [Fortnow and Santhanam, ECCC Report TR07-096] we improve a result of this paper and show for any parameterized problem with a ``linear weak OR'' and with NP-hard underlying classical problem that there is no polynomial reduction from the problem to itself that assigns to ... more >>>

TR08-063 | 21st April 2008
Müller Moritz

Valiant-Vazirani Lemmata for Various Logics

We show analogues of a theorem due to Valiant and Vazirani
for intractable parameterized complexity classes such as W[P], W[SAT]
and the classes of the W-hierarchy as well as those of the A-hierarchy.
We do so by proving a general ``logical'' version of it which may be of
independent interest

... more >>>

TR08-083 | 10th June 2008
Yijia Chen, Jörg Flum

A logic for PTIME and a parameterized halting problem

In [Blass, Gurevich, and Shelah, 99] a logic L_Y has been introduced as a possible candidate for a logic capturing the PTIME properties of structures (even in the absence of an ordering of their universe). A reformulation of this problem in terms of a parameterized halting problem p-Acc for nondeterministic ... more >>>

TR09-131 | 2nd December 2009
Stephan Kreutzer, Anuj Dawar

Parameterized Complexity of First-Order Logic

Revisions: 2

We show that if $\mathcal C$ is a class of graphs which is
"nowhere dense" then first-order model-checking is
fixed-parameter tractable on $\mathcal C$. As all graph classes which exclude a fixed minor, or are of bounded local tree-width or locally exclude a minor are nowhere dense, this generalises algorithmic ... more >>>

TR09-147 | 30th December 2009
Stephan Kreutzer

Algorithmic Meta-Theorems

Algorithmic meta-theorems are general algorithmic results applying to a whole range of problems, rather than just to a single problem alone. They often have a logical and a
structural component, that is they are results of the form:
"every computational problem that can be formalised in a given logic L ... more >>>

TR10-038 | 10th March 2010
Dieter van Melkebeek, Holger Dell

Satisfiability Allows No Nontrivial Sparsification Unless The Polynomial-Time Hierarchy Collapses

Revisions: 1

Consider the following two-player communication process to decide a language $L$: The first player holds the entire input $x$ but is polynomially bounded; the second player is computationally unbounded but does not know any part of $x$; their goal is to cooperatively decide whether $x$ belongs to $L$ at small ... more >>>

TR10-059 | 8th April 2010
Olaf Beyersdorff, Nicola Galesi, Massimo Lauria

Hardness of Parameterized Resolution

Parameterized Resolution and, moreover, a general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider (FOCS'07). In that paper, Dantchev et al. show a complexity gap in tree-like Parameterized Resolution for propositional formulas arising from translations of first-order principles.
We broadly investigate Parameterized Resolution obtaining the following ... more >>>

TR10-198 | 13th December 2010
Olaf Beyersdorff, Nicola Galesi, Massimo Lauria, Alexander Razborov

Parameterized Bounded-Depth Frege is Not Optimal

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider (FOCS'07). In that framework the parameterized version of any proof system is not fpt-bounded for some technical reasons, but we remark that this question becomes much more interesting if we restrict ourselves to those parameterized contradictions ... more >>>

TR11-071 | 27th April 2011
Serge Gaspers, Stefan Szeider

The Parameterized Complexity of Local Consistency

Revisions: 1

We investigate the parameterized complexity of deciding whether a constraint network is $k$-consistent. We show that, parameterized by $k$, the problem is complete for the complexity class co-W[2]. As secondary parameters we consider the maximum domain size $d$ and the maximum number $\ell$ of constraints in which a variable occurs. ... more >>>

TR11-072 | 1st May 2011
Danny Hermelin, Xi Wu

Weak Compositions and Their Applications to Polynomial Lower-Bounds for Kernelization

Revisions: 1

We introduce a new form of composition called \emph{weak composition} that allows us to obtain polynomial kernelization lower-bounds for several natural parameterized problems. Let $d \ge 2$ be some constant and let $L_1, L_2 \subseteq \{0,1\}^* \times \N$ be two parameterized problems where the unparameterized version of $L_1$ is \NP-hard. ... more >>>

TR13-162 | 1st October 2013
Janka Chlebíková, Morgan Chopin

The Firefighter Problem: A Structural Analysis

Revisions: 1

We consider the complexity of the firefighter problem where ${b \geq 1}$ firefighters are available at each time step. This problem is proved NP-complete even on trees of degree at most three and budget one (Finbow et al. 2007) and on trees of bounded degree $b+3$ for any fixed budget ... more >>>

TR14-004 | 30th November 2013
Hasan Abasi, Nader Bshouty, Ariel Gabizon, Elad Haramaty

On $r$-Simple $k$-Path

An $r$-simple $k$-path is a {path} in the graph of length $k$ that
passes through each vertex at most $r$ times. The \simpath{r}{k}
problem, given a graph $G$ as input, asks whether there exists an
$r$-simple $k$-path in $G$. We first show that this problem is
NP-Complete. We then show ... more >>>

TR14-075 | 16th May 2014
Holger Dell

A simple proof that AND-compression of NP-complete problems is hard

Revisions: 3

Drucker (2012) proved the following result: Unless the unlikely complexity-theoretic collapse coNP is in NP/poly occurs, there is no AND-compression for SAT. The result has implications for the compressibility and kernelizability of a whole range of NP-complete parameterized problems. We present a simple proof of this result.

An AND-compression is ... more >>>

TR14-096 | 29th July 2014
Vikraman Arvind, Sebastian Kuhnert, Johannes Köbler, Jacobo Toran

Solving Linear Equations Parameterized by Hamming Weight

Given a system of linear equations $Ax=b$ over the binary field $\mathbb{F}_2$ and an integer $t\ge 1$, we study the following three algorithmic problems:
1. Does $Ax=b$ have a solution of weight at most $t$?
2. Does $Ax=b$ have a solution of weight exactly $t$?
3. Does $Ax=b$ have a ... more >>>

TR14-116 | 6th September 2014
Rahul Mehta

2048 is (PSPACE) Hard, but Sometimes Easy

We prove that a variant of 2048, a popular online puzzle game, is PSPACE-Complete. Our hardness result
holds for a version of the problem where the player has oracle access to the computer player's moves.
Specifically, we show that for an $n \times n$ game board $G$, computing a
more >>>

TR14-134 | 10th October 2014
Martin Lück, Arne Meier, Irina Schindler

Parameterized Complexity of CTL: Courcelle's Theorem For Infinite Vocabularies

Revisions: 3

We present a complete classification of the parameterized complexity of all operator fragments of the satisfiability problem in computation tree logic CTL. The investigated parameterization is temporal depth and pathwidth. Our results show a dichotomy between W[1]-hard and fixed-parameter tractable fragments. The two real operator fragments which are in FPT ... more >>>

TR15-013 | 28th January 2015
Subhash Khot, Igor Shinkar

On Hardness of Approximating the Parameterized Clique Problem

In the $Gap-clique(k, \frac{k}{2})$ problem, the input is an $n$-vertex graph $G$, and the goal is to decide whether $G$ contains a clique of size $k$ or contains no clique of size $\frac{k}{2}$. It is an open question in the study of fixed parameterized tractability whether the $Gap-clique(k, \frac{k}{2})$ problem ... more >>>

TR15-130 | 11th August 2015
Ronald de Haan

An Overview of Non-Uniform Parameterized Complexity

We consider several non-uniform variants of parameterized complexity classes that have been considered in the literature. We do so in a homogenous notation, allowing a clear comparison of the various variants. Additionally, we consider some novel (non-uniform) parameterized complexity classes that come up in the framework of parameterized knowledge compilation. ... more >>>

TR16-157 | 13th October 2016
Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, Jacobo Toran

Parameterized Complexity of Small Weight Automorphisms

We consider the PermCode problem to decide, given a representation of a permutation group G and a parameter k, whether there is a non-trivial element of G with support at most k. This problem generalizes several problems in the literature. We introduce a new method that allows to reduce the ... more >>>

TR17-186 | 29th November 2017
Karthik C. S., Bundit Laekhanukit, Pasin Manurangsi

On the Parameterized Complexity of Approximating Dominating Set

We study the parameterized complexity of approximating the $k$-Dominating Set (domset) problem where an integer $k$ and a graph $G$ on $n$ vertices are given as input, and the goal is to find a dominating set of size at most $F(k) \cdot k$ whenever the graph $G$ has a dominating ... more >>>

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