In resolving instances of a computational problem, if multiple instances of interest share a feature in common, it may be fruitful to compile this feature into a format that allows for more efficient resolution, even if the compilation is relatively expensive. In this article, we introduce a formal framework for classifying problems according to their compilability. The basic object in our framework is that of a parameterized problem, which here is a language along with a parameterization---a map which provides, for each instance, a so-called parameter on which compilation may be performed. Our framework is positioned within the paradigm of parameterized complexity, and our notions are relatable to established concepts in the theory of parameterized complexity. Indeed, we view our framework as playing a unifying role, integrating together parameterized complexity and compilability theory.
We show that the $b$-Coloring problem is complete for the class XNLP when parameterized by the pathwidth of the input graph. Besides determining the precise parameterized complexity of this problem, this implies that b-Coloring parameterized by pathwidth is $W[t]$-hard for all $t$, and resolves the parameterized complexity of $b$-Coloring parameterized by treewidth.
Vertex Cover Problem Parameterized Above and Below Tight Bounds
我们研究了著名的顶点覆盖问题的参数上下紧界。我们证明了其中两个参数化（都是由Mahajan，Raman和Sikdar，J. Computer and System Sciences，75（2）：137- 153，2009提出的）是固定参数易处理的，另外两个参数化是W[1]-hard的（其中一个实际上是W[2]-hard的）。
We study the parameterized complexity of the $s$-Club Cluster Edge Deletion problem: Given a graph $G$ and two integers $s \ge 2$ and $k \ge 1$, is it possible to remove at most $k$ edges from $G$ such that each connected component of the resulting graph has diameter at most $s$? This problem is known to be NP-hard already when $s = 2$. We prove that it admits a fixed-parameter tractable algorithm when parameterized by $s$ and the treewidth of the input graph.
The survey provides an overview of the developing area of parameterized algorithms for graph modification problems. We concentrate on edge modification problems, where the task is to change a small number of adjacencies in a graph in order to satisfy some required property.
A defensive alliance in an undirected graph $G=(V,E)$ is a non-empty set of vertices $S$ satisfying the condition that every vertex $v\in S$ has at least as many neighbours (including itself) in $S$ as it has in $V\setminus S$. We consider the notion of global minimality in this paper. We are interested in globally minimal defensive alliance of maximum size. This problem is known to be NP-hard but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that the Globally Minimal Defensive Alliance…
A set $S$ of vertices of a graph is a defensive alliance if, for each element of $S$, the majority of its neighbours is in $S$. We consider the notion of local minimality in this paper. We are interested in locally minimal defensive alliance of maximum size. We also look at connected version of defensive alliance. This problem is known to be NP-hard but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity. The main results of the paper are the following: (1) Locally Minimal Defensive Alliance is NP-compl…
Parameterized complexity theory was developed in the 1990s to enrich the complexity-theoretic analysis of problems that depend on a range of parameters. In this paper we establish a quantum equivalent of classical parameterized complexity theory, motivated by the need for new tools for the classifications of the complexity of real-world problems. We introduce the quantum analogues of a range of parameterized complexity classes and examine the relationship between these classes, their classical counterparts, and well-studied problems. This framework exposes a rich classification of the complexi…
Machine scheduling problems are a long-time key domain of algorithms and complexity research. A novel approach to machine scheduling problems are fixed-parameter algorithms. To stimulate this thriving research direction, we propose 15 open questions in this area whose resolution we expect to lead to the discovery of new approaches and techniques both in scheduling and parameterized complexity theory.
For every fixed integer $k \geq 1$, we prove that $k$-Edge Colouring is fixed-parameter-tractable when parameterized by the number of vertices of maximum degree.
We define a theory of parameterized algebraic complexity classes in analogy to parameterized Boolean counting classes. We define the classes VFPT and VW[t], which mirror the Boolean counting classes #FPT and #W[t], and define appropriate reductions and completeness notions. Our main contribution is the VW[1]-completeness proof of the parameterized clique family. This proof is far more complicated than in the Boolean world. It requires some new concepts like composition theorems for bounded exponential sums and Boolean-arithmetic formulas. In addition, we also look at two polynomials linked to …
In this paper we study the problem of finding a small safe set $S$ in a graph $G$, i.e. a non-empty set of vertices such that no connected component of $G[S]$ is adjacent to a larger component in $G - S$. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[2]-hard when parameterized by the pathwidth $pw$ and cannot be solved in time $n^{o(pw)}$ unless the ETH is false, (2) it admits no polynomial kernel parameterized by the vertex cover number $vc$ unless $\mathrm{PH} = Σ^{\mathrm{p}}_{3}$, but (3) it is fixed-paramete…
We introduce some classical complexity-theoretic techniques to Parameterized Complexity. First, we study relativization for the machine models that were used by Chen, Flum, and Grohe (2005) to characterize a number of parameterized complexity classes. Here we obtain a new and non-trivial characterization of the A-Hierarchy in terms of oracle machines, and parameterize a famous result of Baker, Gill, and Solovay (1975), by proving that, relative to specific oracles, FPT and A[1] can either coincide or differ (a similar statement holds for FPT and W[P]). Second, we initiate the study of interact…
Flum and Grohe define a parameter (parameterization) as a function $κ$ which maps words over a given alphabet to natural numbers. They require such functions to be polynomial-time computable. We show how this technical restriction can be lifted without breaking the theory. More specifically, instead of $κ$ we consider the set of languages that it bounds as parameterization and define the basic notions of parameterized complexity in terms of promise problems, which completely replace slices. One advantage of this formalization is that it becomes possible to interpret any complexity-theoretic co…
We study the parameterized complexity of the problem to decide whether a given natural number $n$ satisfies a given $Δ_0$-formula $φ(x)$; the parameter is the size of $φ$. This parameterization focusses attention on instances where $n$ is large compared to the size of $φ$. We show unconditionally that this problem does not belong to the parameterized analogue of $\mathsf{AC}^0$. From this we derive that certain natural upper bounds on the complexity of our parameterized problem imply certain separations of classical complexity classes. This connection is obtained via an analysis of a parameter…
We investigate the parameterized complexity of several problems formalizing cluster identification in graphs. In other words we ask whether a graph contains a large enough and sufficiently connected subgraph. We study here three relaxations of CLIQUE: $s$-CLUB and $s$-CLIQUE, in which the relaxation is focused on the distances in respectively the cluster and the original graph, and $γ$-COMPLETE SUBGRAPH in which the relaxation is made on the minimal degree in the cluster. As these three problems are known to be NP-hard, we study here their parameterized complexities. We prove that $s$-CLUB and…
Finding a cycle of lowest weight that represents a homology class in a simplicial complex is known as homology localization (HL). Here we address this NP-complete problem using parameterized complexity theory. We show that it is W[1]-hard to approximate the HL problem when it is parameterized by solution size. We have also designed and implemented two algorithms based on treewidth solving the HL problem in FPT-time. Both algorithms are ETH-tight but our results shows that one outperforms the other in practice.
Deciding whether a diagram of a knot can be untangled with a given number of moves (as a part of the input) is known to be NP-complete. In this paper we determine the parameterized complexity of this problem with respect to a natural parameter called defect. Roughly speaking, it measures the efficiency of the moves used in the shortest untangling sequence of Reidemeister moves. We show that the II- moves in a shortest untangling sequence can be essentially performed greedily. Using that, we show that this problem belongs to W[P] when parameterized by the defect. We also show that this problem …
