Jonas Bamse Andersen, Jørgen Bang-Jensen, Anders Yeo
We give an FPT algorithm for deciding whether the vertex set a digraph $D$ can be partitioned into two disjoint sets $V_1,V_2$ such that the digraph $D[V_1]$ induced by $V_1$ has a vertex that can reach all other vertices by directed paths, the digraph $D[V_2]$ has no vertex of in-degree zero and $|V_i|\geq k_i$, where $k_1,k_2$ are part of the input. This settles an open problem from[1,4].
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A Proof Checking View of Parameterized Complexity
PCP定理是计算复杂性理论中最令人震惊的结果之一，是一系列关于证明检验的结果的顶点，它揭示了计算问题的一些深层结构。作为一个令人惊讶的副作用，它也给出了强大的非近似性结果。在本文中，我们发起的证明检查的参数化复杂性范围内的研究。特别是，我们适应和扩展PCP[n log log n，n log log n] Feige等人的结果。几个参数化类，并讨论了一些推论。
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…
Parameterized Control Complexity in Fallback Voting
我们研究了参数化控制的复杂性回退投票，一个投票系统，结合了基于偏好的批准投票。选举控制是外部代理人篡改选举结果的许多不同方式之一。我们证明了在后备投票中添加和删除候选人对于建设性和破坏性的情况都是W[2]-困难的，由外部代理所采取的行动量来参数化。此外，我们表明，增加和删除选民在后备投票是W[2]-难的建设性的情况下，参数化的外部代理所采取的行动的数量，并在FPT的破坏性的情况下。
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…
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…
Every square matrix $A=(a_{uv})\in \mathcal{C}^{n\times n}$ can be represented as a digraph having $n$ vertices. In the digraph, a block (or 2-connected component) is a maximally connected subdigraph that has no cut-vertex. The determinant and the permanent of a matrix can be calculated in terms of the determinant and the permanent of some specific induced subdigraphs of the blocks in the digraph. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Such partitions of the digraph are called the $\mathcal{B}$-partitions. In this paper, first, we develop a…
On the Parameterized Complexity of the Perfect Phylogeny Problem
本文分别以基因数和颜色数为参数，证明了Perfect Phylogeny和Triangulating Colored Graphs问题属于XALP完全类，并在指数时间假设下证明了不存在$f(k) n^{o(k)}$时间的求解算法。
We study parameterized Constraint Satisfaction Problem for infinite constraint languages. The parameters that we study are weight of the satisfying assignment, number of constraints, maximum number of occurrences of a variable in the instance, and maximum number of occurrences of a variable in each constraint. A dichotomy theorem is already known for finite constraint languages with the weight parameter. We prove some general theorems that show, as new results, that some well-known problems are fixed-parameter tractable and some others are in W[1].
Completeness Results for Parameterized Space Classes
一旦证明问题位于FPT中或对于W层次结构或类似参数化层次结构中的类是完整的，则该问题的参数化复杂性被认为是“已解决”的。然而，一些自然参数化的问题，抵制这样的分类。至少在某些情况下，原因是最近获得了参数化空间复杂度的上界和下界，这排除了参数化时间类的完整性结果。在本文中，我们在这方面取得了进展，证明了结合生成性问题和最长公共子序列问题是完全的参数化空间类。这些类定义在不同形式的有界非决定论和同时的时间-空间界限。作为一种技术工具，我们引入了一个“联盟操作”，经典的复杂性类和W-类之间的问题完成转换。
We present completeness results for inference in Bayesian networks with respect to two different parameterizations, namely the number of variables and the topological vertex separation number. For this we introduce the parameterized complexity classes $\mathsf{W[1]PP}$ and $\mathsf{XLPP}$, which relate to $\mathsf{W[1]}$ and $\mathsf{XNLP}$ respectively as $\mathsf{PP}$ does to $\mathsf{NP}$. The second parameter is intended as a natural translation of the notion of pathwidth to the case of directed acyclic graphs, and as such it is a stronger parameter than the more commonly considered treewi…
Parameterized complexity in multiple-interval graphs: domination, partition, separation, irredundancy
我们证明了问题k-支配集及其几个变体，包括k-连通支配集、k-独立支配集和k-配团，当以解大小k为参数时，在多区间图或其补图或两者中都是W[1]-困难的.另一方面，我们证明了当限制为多区间图及其补图时，这些问题属于W[1]。这回答了Fellows等人的一个公开问题。与之形成鲜明对比的是，我们证明了在多区间图及其补图中，d >= 2的d-距离k-支配集是W[2]-完全的。我们还证明了k-完美码和d-距离k-完美码（d >= 2）即使在单位2-道区间图中也是W[1]-完全的。此外，我们还给出了多区间图及其补图中k-点团划分和k-分离点的参数化复杂性的各种新结果，并给出了一般图中k-无冗余集的W[1]-硬度的一个非常简单的替代证明。
Parameterized Proof Complexity and W[1]
我们启动了一个参数化证明复杂性的程序，旨在提供FPT不同于W[1]的证据。类似的程序已经存在于W[2]和W[SAT]类中。我们比较了这些程序，并证明了W[1]-参数化Resolution的上下界。
The embedding problem is to decide, given an ordered pair of structures, whether or not there is an injective homomorphism from the first structure to the second. We study this problem using an established perspective in parameterized complexity theory: the universe size of the first structure is taken to be the parameter, and we define the embedding problem relative to a class ${\cl A}$ of structures to be the restricted version of the general problem where the first structure must come from ${\cl A}$. We initiate a systematic complexity study of this problem family, by considering classes wh…
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 consider the parameterized complexity of the following scheduling problem. We must schedule a number of jobs on $m$ machines, where each job has unit length, and the graph of precedence constraints consists of a set of chains. Each precedence constraint is labelled with an integer that denotes the exact (or minimum) delay between the jobs. We study different cases; delays can be given in unary and in binary, and the case that we have a single machine is discussed separately. We consider the complexity of this problem parameterized by the number of chains, and by the thickness…
Parameterized Complexity of Graph Constraint Logic
图约束逻辑是由Hearn和Demaine引入的框架，它提供了几个问题，这些问题通常是简化的方便起点。我们研究了约束图可满足性的参数化复杂性和有界和无界版本的非确定性约束逻辑（NCL）的解决方案的长度，树宽和最大程度的基本约束图作为参数。作为一个主要的结果，我们表明，限制NCL保持PSPACE-完全图的有界带宽，加强赫恩和Demaine的框架。这使我们能够改善现有的结果，从NCL减少。我们证明了几个经典的图问题（包括独立集，反馈顶点集和支配集）的重构版本是PSPACE-完全的有界带宽的平面图和高峰时间，推广到$k\times n$板，是PSPACE-完全的，即使当$k$是最多一个常数。
We study the NP-complete Minimum Shared Edges (MSE) problem. Given an undirected graph, a source and a sink vertex, and two integers p and k, the question is whether there are p paths in the graph connecting the source with the sink and sharing at most k edges. Herein, an edge is shared if it appears in at least two paths. We show that MSE is W[1]-hard when parameterized by the treewidth of the input graph and the number k of shared edges combined. We show that MSE is fixed-parameter tractable with respect to p, but does not admit a polynomial-size kernel (unless NP is contained in coNP/poly).…
Parameterized Complexity of Edge-Constrained Graph Partitioning
该研究分析了边约束图划分（ECGP）及其平衡、带符号变体的NP难性、多项式核、FPT算法与W[1]难性等参数化复杂性结果。
We demonstrate some lower bounds for parameterized problems via parameterized classes corresponding to the classical ${\rm AC}^0$. Among others, we derive such a lower bound for all fpt-approximations of the parameterized clique problem and for a parameterized halting problem, which recently turned out to link problems of computational complexity, descriptive complexity, and proof theory. To show the first lower bound, we prove a strong ${\rm AC}^0$ version of the planted clique conjecture: ${\rm AC}^0$-circuits asymptotically almost surely can not distinguish between a random graph and this g…
The Satisfactory Partition problem consists in deciding if the set of vertices of a given undirected graph can be partitioned into two nonempty parts such that each vertex has at least as many neighbours in its part as in the other part. This problem was introduced by Gerber and Kobler [European J. Oper. Res. 125 (2000) 283-291] and further studied by other authors, but its parameterized complexity remains open until now. It is known that the Satisfactory Partition problem, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. We enhance our understa…
Let $G$ be a graph on $n$ vertices and $\mathrm{STAB}_k(G)$ be the convex hull of characteristic vectors of its independent sets of size at most $k$. We study extension complexity of $\mathrm{STAB}_k(G)$ with respect to a fixed parameter $k$ (analogously to, e.g., parameterized computational complexity of problems). We show that for graphs $G$ from a class of bounded expansion it holds that $\mathrm{xc}(\mathrm{STAB}_k(G))\leqslant \mathcal{O}(f(k)\cdot n)$ where the function $f$ depends only on the class. This result can be extended in a simple way to a wide range of similarly defined graph p…
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…
We study the parameterized complexity of several positional games. Our main result is that Short Generalized Hex is W[1]-complete parameterized by the number of moves. This solves an open problem from Downey and Fellows' influential list of open problems from 1999. Previously, the problem was thought of as a natural candidate for AW[*]-completeness. Our main tool is a new fragment of first-order logic where universally quantified variables only occur in inequalities. We show that model-checking on arbitrary relational structures for a formula in this fragment is W[1]-complete when parameterize…
Fantastic Flips and Where to Find Them: A General Framework for Parameterized Local Search on Partitioning Problem
针对划分问题，提出了一种结合局部搜索与参数化算法的通用框架，通过引入类型数参数τ，实现了在τ^k 2^{O(k)} |I|^{O(1)}时间内的求解，并证明了该算法在ETH假设下的最优性及问题在仅以k为参数时的W[1]-困难性。
An $(r, \ell)$-partition of a graph $G$ is a partition of its vertex set into $r$ independent sets and $\ell$ cliques. A graph is $(r, \ell)$ if it admits an $(r, \ell)$-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is $(r,\ell)$-well-covered if it is both $(r,\ell)$ and well-covered. In this paper we consider two different decision problems. In the $(r,\ell)$-Well-Covered Graph problem ($(r,\ell)$WCG for short), we are given a graph $G$, and the question is whether $G$ is an $(r,\ell)$-well-covered graph. In the Well-Covered $(r,\ell)$-Graph prob…
A Parameterized-Complexity Framework for Finding Local Optima
本文提出了一种新的参数复杂度框架，用于寻找局部最优解，通过结合参数复杂度范式，实现了强下界和可处理性结果。
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.
Parameterized Complexity Results in Symmetry Breaking
对称性是许多组合问题的共同特征。不幸的是，从一个问题中消除所有的对称性在计算上往往是困难的。本文认为，最近的参数化复杂性的结果提供了深入了解，棘手的，并帮助确定特殊情况下，对称性可以处理得更容易
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.
In this article, we study the complexity of weighted team definability for logics with team semantics. This problem is a natural analogue of one of the most studied problems in parameterized complexity, the notion of weighted Fagin-definability, which is formulated in terms of satisfaction of first-order formulas with free relation variables. We focus on the parameterized complexity of weighted team definability for a fixed formula phi of central team-based logics. Given a first-order structure A and the parameter value k as input, the question is to determine whether A,T models phi for some t…
Computational Social Choice: Parameterized Complexity and Challenges
该研究综述了计算社会选择领域的多赢家确定、享乐博弈两大关键问题，聚焦其参数化复杂度，并提出了相关研究挑战。
On the Average-case Complexity of Parameterized Clique
k-团问题是一个基本的组合问题，在经典复杂性理论和参数化复杂性理论中起着重要作用。它是最著名的NP-完全和W[1]-完全问题之一。此外，自20世纪70年代以来，它的平均情况复杂性分析已经创造了一个很长的研究线索。在这里，我们通过研究k-团问题的平均情况复杂度对参数k的依赖性来继续这一研究路线。为此，我们定义了两个自然的参数化类似物的有效的平均情况下的算法。然后，我们表明，k-团承认这两个类似的任意密度的Erdens-Rényi随机图。我们还表明，k-团是不太可能承认这些类似物的一些特定的可计算的输入分布。
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.
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.
Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.
Description logics are knowledge representation languages that have been designed to strike a balance between expressivity and computational tractability. Many different description logics have been developed, and numerous computational problems for these logics have been studied for their computational complexity. However, essentially all complexity analyses of reasoning problems for description logics use the one-dimensional framework of classical complexity theory. The multi-dimensional framework of parameterized complexity theory is able to provide a much more detailed image of the complex…
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.
The Parameterized Complexity of Global Constraints
我们认为，参数化的复杂性是一个有用的工具，研究全球的约束。特别是，我们证明了许多难以完全传播的全局约束具有自然参数，这使得它们具有易于计算的固定参数。这种易处理性往往是一个简单的动态程序或分解的结果，它有一个强大的后门的大小有界。这个强后门通常是一个循环割集。我们还表明，参数化的复杂性可以用来研究其他方面的约束规划，如对称破缺。例如，我们证明了值对称性是固定参数易破的对称性的数量。最后，我们认为，参数化的复杂性可以用来推导出的约束传播的近似性的结果。
The Fine Classification of Conjunctive Queries and Parameterized Logarithmic Space Complexity
我们从参数化复杂性的角度对合取查询计算的复杂性进行了一个基本的研究。我们根据问题的复杂性对布尔合取查询集进行分类。以前的工作表明，一组合取查询是固定参数易处理的，正是当一组等价于一组查询有界树宽。我们提出了一个很好的分类查询集参数化对数空间减少。我们表明，在有界树宽制度，有三个复杂度，并确定查询集的程度有界的路径宽度和有界树深度的属性。我们还通过对数空间机器刻画和完备问题研究了两个更高的度。我们的工作产生了一个显着丰富的角度对合取查询的复杂性，并在同一时间，提出了新的途径，参数化的复杂性的研究。
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…
Preprocessing Complexity for Some Graph Problems Parameterized by Structural Parameters
针对多个基本图问题，以结构参数为参数，利用多种技术完整刻画了其预处理复杂性，填补了多项式核存在性研究的空白。
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.
Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction
本文研究了biclique收缩和平衡biclique收缩的参数复杂性，证明了其NP难性，并提出了FPT算法和内核压缩方法。
We consider the problems of deciding whether an input graph can be modified by removing/adding at most k vertices/edges such that the result of the modification satisfies some property definable in first-order logic. We establish a number of sufficient and necessary conditions on the quantification pattern of the first-order formula ϕfor the problem to be fixed-parameter tractable or to admit a polynomial kernel.
We complement the recent algorithmic result that Feedback Vertex Set is XP-time solvable parameterized by the mim-width of a given branch decomposition of the input graph [3] by showing that the problem is W[1]-hard in this parameterization. The hardness holds even for linear mim-width, as well as for H-graphs, where the parameter is the number of edges in H. To obtain this result, we adapt a reduction due to Fomin, Golovach and Raymond [2], following the same line of reasoning but adding a new gadget.
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…
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 study the problem of exchange when 1) agents are endowed with heterogeneous indivisible objects, and 2) there is no money. In general, no rule satisfies the three central properties Pareto-efficiency, individual rationality, and strategy-proofness \cite{Sonmez1999}. Recently, it was shown that Top Trading Cycles is $\NP$-hard to manipulate \cite{FujitaEA2015}, a relaxation of strategy-proofness. However, parameterized complexity is a more appropriate framework for this and other economic settings. Certain aspects of the problem - number of objects each agent brings to the table, goods up fo…
Parameterized Complexity Results for Plan Reuse
规划是一个众所周知的高最坏情况复杂性的计算困难的问题。研究人员一直在投入大量的精力来开发规划或限制，使规划实际可行。基于案例的规划是一种启发式方法，在解决类似问题时，人们试图重用以前的经验，以避免一些规划工作。在某些情况下，计划重用可能会为计划生成提供一种有趣的替代方案。我们提供的理论结果，确定计划重用的情况下，可证明是听话的。我们在参数化复杂性的框架中进行分析，该框架支持严格的最坏情况复杂性分析，该分析将输入的结构特性考虑在内。参数化复杂性的一个中心概念是固定参数可处理性，它通过利用问题输入的结构属性的影响来扩展多项式时间可处理性的经典概念。我们绘制了一个详细的地图的参数化的复杂性景观的几个变种的问题，出现在基于案例的规划的背景下。特别是，我们考虑重用现有计划的问题，在参数方面施加各种限制，例如可以添加到现有计划的步骤数量，将其变成手头的规划实例的解决方案。
On the Parameterized Complexity of Reconfiguration Problems
我们提出的第一个结果的参数化的复杂性的重新配置问题，其中一个重新配置版本的优化问题$Q$作为输入两个可行的解决方案$S$和$T$，并确定是否有一个序列的{ \em重新配置步骤}，可以应用到转换$S$到$T$，使每个步骤的结果在一个可行的解决方案，以$Q$。对于本文中的大多数结果，$S$和$T$是给定图的顶点的子集，并且重新配置步骤添加或删除顶点。我们的研究的动机是最近的结果，建立大多数NP-难的问题，经典的复杂性的重构是PSPACE完全的。我们解决的问题下两个自然参数化的几个重要的图形属性：$k$，解决方案的大小，和$\ell$，步骤序列的长度。我们的第一个一般结果是一个算法范例，{ \em重新配置内核}，用于获得固定参数算法的重新配置版本的{ \sc顶点覆盖}，更一般地说，{ \sc有界命中集}和{ \sc反馈顶点集}，所有参数由$k$。相反，我们表明，重新配置{ \sc无界命中集}是$W[2]$ -硬参数时，$k+\ell$。我们还证明了$W[1]$ -硬度的重构版本的一大类最大化问题参数化的$k+\ell$，其相应的删除问题参数化的$\ell$，在这样做，我们表明，FPT中存在的问题时，参数化的$k$，但其重构版本是$W[1]$ -硬参数化时，由$k+\ell$。
A Parameterized Theory of PAC Learning
本文提出参数化PAC学习理论，定义两种对应FPT的固定参数可学习性概念及不可学习性证明工具，并刻画CNF、DNF与图学习问题的精细可处理边界。
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 …
Parameterized Complexity of CTL: A Generalization of Courcelle's Theorem
我们提出了一个几乎完整的分类的可满足性问题的计算树逻辑CTL的所有操作符片段的参数化复杂性。研究的参数化是时间深度和结构路径宽度的总和。分类显示了W[1]-硬和固定参数易处理片段之间的二分法。FPT中唯一被证实的真正的操作符片段是仅包含AX的片段。此外，我们证明了一个推广的Courcelle定理的无限签名，这将被用来证明FPT成员的情况。
The Parameterized Complexity of Quantum Verification
本文研究以非 Clifford 门数量为参数的 QMA 问题复杂度，证明参数化量子电路可满足性问题存在运行时间随非 Clifford 门数指数增长、随系统规模多项式增长的经典算法，并给出 T 计数下界及量子非恒等性检验的参数化复杂度分析。
Paradigms for Parameterized Enumeration
本文的目的是从参数化复杂性的角度研究枚举的计算复杂性和算法性。枚举是输出给定问题的所有解的任务。首先，我们在参数化复杂性的背景下正式定义不同的有效枚举概念。其次，我们展示了如何不同的算法范例可以使用，以获得参数有效的枚举算法在一些例子。这些范例使用来自参数化决策设计的众所周知的原则以及枚举技术，例如内核化和自归约。核化的概念，特别是，导致固定参数易处理的枚举问题的表征。
Parameterized complexity allows us to analyze the time complexity of problems with respect to a natural parameter depending on the problem. Reoptimization looks for solutions or approximations for problem instances when given solutions to neighboring instances. We try to combine both techniques, in order to better classify the complexity of problems in the parameterized setting. Specifically, we see that some problems in the class of compositional problems, which do not have polynomial kernels under standard complexity-theoretic assumptions, do have polynomial kernels under reoptimization for …
The parameterized complexity of some geometric problems in unbounded dimension
我们研究了以下基本几何问题关于维数$d$的参数化复杂性：i）给定$\Rd$中的$n$点，计算它们的最小包围柱。ii）给定$\Rd$中的两个$n$ -点集，判断它们是否可以被两个超平面分开。iii）给定一个含$d$变量的$n$线性不等式组，求其最大可行子系统。我们证明了当用维数$d$参数化时，所有这些问题的决策版本都是W[1]-困难的. %，因此对于任何可计算函数$f$和常数$c$ %（除非FPT=W[1]），在${O}(f(d)n^c)$时间内不可解。我们的约化也给出了一个$n^{Ω(d)}$ -时间下界（在指数时间假设下）.
We prove that for every class $C$ of graphs with effectively bounded expansion, given a first-order sentence $φ$ and an $n$-element structure $\mathbb{A}$ whose Gaifman graph belongs to $C$, the question whether $φ$ holds in $\mathbb{A}$ can be decided by a family of AC-circuits of size $f(φ)\cdot n^c$ and depth $f(φ)+c\log n$, where $f$ is a computable function and $c$ is a universal constant. This places the model-checking problem for classes of bounded expansion in the parameterized circuit complexity class $para-AC^1$. On the route to our result we prove that the basic decomposition toolbo…
For two integers $r, \ell \geq 0$, a graph $G = (V, E)$ is an $(r,\ell)$-graph if $V$ can be partitioned into $r$ independent sets and $\ell$ cliques. In the parameterized $(r,\ell)$-Vertex Deletion problem, given a graph $G$ and an integer $k$, one has to decide whether at most $k$ vertices can be removed from $G$ to obtain an $(r,\ell)$-graph. This problem is NP-hard if $r+\ell \geq 1$ and encompasses several relevant problems such as Vertex Cover and Odd Cycle Transversal. The parameterized complexity of $(r,\ell)$-Vertex Deletion was known for all values of $(r,\ell)$ except for $(2,1)$, $…
Parameterized Complexity of Caching in Networks
本文通过参数化复杂度范式对网络缓存问题进行全面分析，填补了除一般NP-hardness外可处理性边界未知的空白，并确立了算法上下界以明确该问题变得可处理的具体条件。
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