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Properties of graphs specified by a regular language 由常规语言指定的图的属性
IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Pub Date : 2022-08-12 DOI: 10.1007/s00236-022-00427-z
Volker Diekert, Henning Fernau, Petra Wolf

Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property (varPhi ). What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there is a graph satisfying (varPhi ) in the family? We approach this question by using formal languages for specifying families of graphs, in particular by regular sets of words. We show that certain graph properties can be decided by studying the syntactic monoid of the specification language L if a certain torsion condition is satisfied. This condition holds trivially if L is regular. More specifically, we use a natural binary encoding of finite graphs over a binary alphabet (varSigma ), and we define a regular set (mathbb {G}subseteq varSigma ^*) such that every nonempty word (win mathbb {G}) defines a finite and nonempty graph. Also, graph properties can then be syntactically defined as languages over (varSigma ). Then, we ask whether the automaton (mathcal {A}) specifies some graph satisfying a certain property (varPhi ). Our structural results show that we can answer this question for all “typical” graph properties. In order to show our results, we split L into a finite union of subsets and every subset of this union defines in a natural way a single finite graph F where some edges and vertices are marked. The marked graph in turn defines an infinite graph (F^infty ) and therefore the family of finite subgraphs of (F^infty ) where F appears as an induced subgraph. This yields a geometric description of all graphs specified by L based on splitting L into finitely many pieces; then using the notion of graph retraction, we obtain an easily understandable description of the graphs in each piece.

传统上,图算法得到一个单一的图作为输入,然后他们应该决定这个图是否满足某个属性(varPhi )。如果这个问题被修改了,我们得到一个可能无限的图族作为输入,问题是是否有一个图族中满足(varPhi ) ?我们通过使用形式语言来指定图族,特别是正则词集来解决这个问题。我们证明了如果满足一定的扭转条件,可以通过研究规范语言L的语法单群来确定某些图的性质。如果L是正则的,这个条件一般成立。更具体地说,我们在二进制字母表(varSigma )上使用有限图的自然二进制编码,并定义一个正则集(mathbb {G}subseteq varSigma ^*),使得每个非空单词(win mathbb {G})定义一个有限和非空图。此外,图形属性可以在语法上定义为(varSigma )上的语言。然后,我们问自动机(mathcal {A})是否指定满足某一属性的某个图(varPhi )。我们的结构结果表明,我们可以对所有“典型”图属性回答这个问题。为了证明我们的结果,我们将L分割成一个子集的有限并集,这个并集的每个子集以自然的方式定义了一个有限图F,其中一些边和顶点被标记。标记的图又定义了一个无限图(F^infty ),因此定义了(F^infty )的有限子图族,其中F作为诱导子图出现。这产生了L所指定的所有图的几何描述,基于将L分成有限多块;然后利用图可收回的概念,我们得到了每个图块中易于理解的图的描述。
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引用次数: 3
Preface to Klaus-Jörn Lange Festschrift 克劳斯-约恩-兰格艺术节序言
IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Pub Date : 2022-08-09 DOI: 10.1007/s00236-022-00426-0
Henning Fernau, Markus Holzer, Petra Wolf
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引用次数: 0
On the decidability of finding a positive ILP-instance in a regular set of ILP-instances 关于在正则ILP实例集中找到正ILP实例的判定性
IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Pub Date : 2022-08-04 DOI: 10.1007/s00236-022-00429-x
Petra Wolf

The regular intersection emptiness problem for a decision problem P (({{textit{int}}_{{mathrm {Reg}}}})(P)) is to decide whether a potentially infinite regular set of encoded P-instances contains a positive one. Since ({{textit{int}}_{{mathrm {Reg}}}})(P) is decidable for some NP-complete problems and undecidable for others, its investigation provides insights in the nature of NP-complete problems. Moreover, the decidability of the ({{textit{int}}_{{mathrm {Reg}}}})-problem is usually achieved by exploiting the regularity of the set of instances; thus, it also establishes a connection to formal language and automata theory. We consider the ({{textit{int}}_{{mathrm {Reg}}}})-problem for the well-known NP-complete problem Integer Linear Programming (ILP). It is shown that any DFA that describes a set of ILP-instances (in a natural encoding) can be reduced to a finite core of instances that contains a positive one if and only if the original set of instances did. This result yields the decidability of ({{textit{int}}_{{mathrm {Reg}}}})(ILP).

决策问题P (({{textit{int}}_{{mathrm {Reg}}}}) (P))的正则相交空性问题是决定编码P-实例的潜在无限正则集是否包含一个正的P-实例。由于({{textit{int}}_{{mathrm {Reg}}}}) (P)对于某些np完全问题是可决定的,而对于其他np完全问题是不可决定的,因此对它的研究提供了对np完全问题本质的见解。此外,({{textit{int}}_{{mathrm {Reg}}}}) -问题的可判定性通常是通过利用实例集的规律性来实现的;因此,它也与形式语言和自动机理论建立了联系。我们考虑了众所周知的np完全问题整数线性规划(ILP)的({{textit{int}}_{{mathrm {Reg}}}}) -问题。证明了描述一组ilp实例(以自然编码)的任何DFA都可以简化为包含正的实例的有限核,当且仅当原始实例集包含正的实例。该结果得到({{textit{int}}_{{mathrm {Reg}}}}) (ILP)的可判定性。
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引用次数: 0
Pushdown automata and constant height: decidability and bounds 下推自动机与常高:可判定性与边界
IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Pub Date : 2022-07-27 DOI: 10.1007/s00236-022-00434-0
Giovanni Pighizzini, Luca Prigioniero

It cannot be decided whether a pushdown automaton accepts using a pushdown height, which does not depend on the input length, i.e., when it accepts using constant height. Furthermore, when a pushdown automaton accepts in constant height, the height can be arbitrarily large with respect to the size of the description of the machine, namely it does not exist any recursive function in the size of the description of the machine bounding the height of the pushdown. In contrast, in the restricted case of pushdown automata over a one-letter input alphabet, i.e., unary pushdown automata, the situation is different. First, acceptance in constant height is decidable. Moreover, in the case of acceptance in constant height, the height is at most exponential with respect to the size of the description of the pushdown automaton. We also prove a matching lower bound. Finally, if a unary pushdown automaton uses nonconstant height to accept, then the height should grow at least as the logarithm of the input length. This bound is optimal.

不能确定下推自动机是否接受使用下推高度,这与输入长度无关,即当它接受使用恒定高度时。此外,当下推自动机接受恒定高度时,高度相对于机器描述的大小可以任意大,即在机器描述的大小中不存在约束下推高度的递归函数。相反,在输入字母为一个字母的下推自动机的限制情况下,即一元下推自动机,情况就不同了。首先,恒定高度的接受度是可以确定的。此外,在接受恒定高度的情况下,高度最多与下推自动机描述的大小成指数关系。我们还证明了一个匹配的下界。最后,如果一元下推自动机使用非恒定高度来接受,那么高度应该至少随着输入长度的对数而增长。这个界是最优的。
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引用次数: 4
The regular languages of wire linear AC0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$^0$$end{document} wire linear AC0documentclass[12pt]{minimal} usepackage{amsmath} userpackage{wasysym}usepackage{amsfonts} use package{amssymb} user package{asbssy}use package{mathrsfs} usapackage{upgeek}setlength{doddsedmargin}{-69pt} begin{document}$^0$end
IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Pub Date : 2022-07-25 DOI: 10.1007/s00236-022-00432-2
M. Cadilhac, Charles Paperman
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引用次数: 0
Reoptimization of parameterized problems 参数化问题的再优化
IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Pub Date : 2022-07-25 DOI: 10.1007/s00236-022-00428-y
Hans-Joachim Böckenhauer, Elisabet Burjons, Martin Raszyk, Peter Rossmanith

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 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 the reoptimization model for some local modifications. We also observe that, for some other local modifications, these same problems do not have polynomial kernels unless (mathbf{NP}subseteq mathbf{coNP/poly}). We find examples of compositional problems, whose reoptimization versions do not have polynomial kernels under any of the considered local modifications. Finally, in another negative result, we prove that the reoptimization version of Connected Vertex Cover does not have a polynomial kernel unless Set Cover has a polynomial compression. In a different direction, looking at problems with polynomial kernels, we find that the reoptimization version of Vertex Cover has a polynomial kernel of size (varvec{2k+1}) using crown decompositions only, which improves the size of the kernel achievable with this technique in the classic problem.

参数化复杂度允许我们根据问题的自然参数来分析问题的时间复杂度。当给定相邻实例的解时,重新优化寻找问题实例的解或近似值。我们将这两种技术结合起来,以便在参数化设置中更好地分类问题的复杂性。具体来说,我们发现在标准复杂性理论假设下不具有多项式核的组合问题,在局部修正的再优化模型下具有多项式核。我们还观察到,对于其他一些局部修正,这些相同的问题没有多项式核,除非(mathbf{NP}subseteq mathbf{coNP/poly})。我们找到了组合问题的例子,这些问题的再优化版本在任何考虑的局部修改下都不具有多项式核。最后,在另一个否定的结果中,我们证明了连通顶点覆盖的再优化版本不具有多项式核,除非集合覆盖具有多项式压缩。在另一个方向上,看看多项式核的问题,我们发现顶点覆盖的再优化版本只有一个大小为(varvec{2k+1})的多项式核,这提高了用这种技术在经典问题中可以实现的核的大小。
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引用次数: 6
The regular languages of wire linear AC(^0) 有线线性AC(^0)的正则语言
IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Pub Date : 2022-07-25 DOI: 10.1007/s00236-022-00432-2
Michaël Cadilhac, Charles Paperman

In this paper, the regular languages of wire linear (hbox {AC}^0)are characterized as the languages expressible in the two-variable fragment of first-order logic with regular predicates, (mathrm{FO}^2[mathrm{reg}]). Additionally, they are characterized as the languages recognized by the algebraic class (mathbf {QLDA}). The class is shown to be decidable and examples of languages in and outside of it are presented.

在本文中,有线线性(hbox{AC}^0)的正则语言被刻画为可在具有正则谓词的一阶逻辑的双变量片段中表达的语言,(mathrm{FO}^2[mathrm{reg}])。此外,它们被描述为代数类(mathbf{QLDA})所识别的语言。该类被证明是可判定的,并给出了类内外语言的例子。
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引用次数: 0
Careful synchronization of partial deterministic finite automata 部分确定性有限自动机的谨慎同步
IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Pub Date : 2022-07-22 DOI: 10.1007/s00236-022-00433-1
Hanan Shabana, M. V. Volkov

We approach the task of computing a carefully synchronizing word of minimum length for a given partial deterministic automaton, encoding the problem as an instance of SAT and invoking a SAT solver. Our experiments demonstrate that this approach gives satisfactory results for automata with up to 100 states even if very modest computational resources are used. We compare our results with the ones obtained by the first author for exact synchronization, which is another version of synchronization studied in the literature, and draw some theoretical conclusions.

我们处理的任务是为给定的部分确定性自动机计算一个最小长度的仔细同步字,将问题编码为SAT实例,并调用SAT求解器。我们的实验表明,即使使用了非常有限的计算资源,这种方法也能为多达100个状态的自动机提供令人满意的结果。我们将我们的结果与第一作者获得的精确同步的结果进行了比较,并得出了一些理论结论。精确同步是文献中研究的同步的另一个版本。
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引用次数: 1
Operational complexity and pumping lemmas 操作复杂性和泵浦引理
IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Pub Date : 2022-07-20 DOI: 10.1007/s00236-022-00431-3
Jürgen Dassow, Ismaël Jecker

The well-known pumping lemma for regular languages states that, for any regular language L, there is a constant p (depending on L) such that the following holds: If (win L) and (vert wvert ge p), then there are words (xin V^{*}), (yin V^+), and (zin V^{*}) such that (w=xyz) and (xy^tzin L) for (tge 0). The minimal pumping constant ({{{,mathrm{mpc},}}(L)}) of L is the minimal number p for which the conditions of the pumping lemma are satisfied. We investigate the behaviour of ({{{,mathrm{mpc},}}}) with respect to operations, i. e., for an n-ary regularity preserving operation (circ ), we study the set ({g_{circ }^{{{,mathrm{mpc},}}}(k_1,k_2,ldots ,k_n)}) of all numbers k such that there are regular languages (L_1,L_2,ldots ,L_n) with ({{{,mathrm{mpc},}}(L_i)=k_i}) for (1le ile n) and ({{{,mathrm{mpc},}}(circ (L_1,L_2,ldots ,L_n)=~k}). With respect to Kleene closure, complement, reversal, prefix and suffix-closure, circular shift, union, intersection, set-subtraction, symmetric difference,and concatenation, we determine ({g_{circ }^{{{,mathrm{mpc},}}}(k_1,k_2,ldots ,k_n)}) completely. Furthermore, we give some results with respect to the minimal pumping length where, in addition, (vert xyvert le p) has to hold.

众所周知的正则语言抽吸引理指出,对于任何正则语言L,都存在一个常数p(取决于L),使得以下条件成立:如果(win L)和(vert wvert ge p),则存在单词(xin V^{*})、(yin V^+)和(zin V^{*}),使得(tge 0)对应(w=xyz)和(xy^tzin L)。L的最小抽运常数({{{,mathrm{mpc},}}(L)})是满足抽运引理条件的最小数p。我们研究了({{{,mathrm{mpc},}}})在操作方面的行为,即,对于一个n元正则性保持操作(circ ),我们研究了所有数字k的集合({g_{circ }^{{{,mathrm{mpc},}}}(k_1,k_2,ldots ,k_n)}),使得(1le ile n)和({{{,mathrm{mpc},}}(circ (L_1,L_2,ldots ,L_n)=~k})都有正则语言(L_1,L_2,ldots ,L_n)和({{{,mathrm{mpc},}}(L_i)=k_i})。对于Kleene闭包、补包、反转、前缀和后缀闭包、圆移位、并并、交集、集减法、对称差分和连接,我们完全确定了({g_{circ }^{{{,mathrm{mpc},}}}(k_1,k_2,ldots ,k_n)})。此外,我们给出了一些关于最小泵浦长度的结果,此外,(vert xyvert le p)必须保持不变。
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引用次数: 1
Weighted throughput in a single machine preemptive scheduling with continuous controllable processing times 具有连续可控处理时间的单机抢占调度中的加权吞吐量
IF 0.6 4区 计算机科学 Q4 COMPUTER SCIENCE, INFORMATION SYSTEMS Pub Date : 2022-07-19 DOI: 10.1007/s00236-022-00430-4
Asaf Levin, Tal Shusterman

We consider the problem of weighted throughput in the single machine preemptive scheduling with continuous controllable processing times. A set of tasks can be scheduled on a single machine. Each task j is associated with a nonnegative weight (w_{j}), a release date, a due date, and an interval of possible processing times. A task j can either be scheduled with a total processing time (p_j) which is in the given interval, or rejected (not participating in the schedule). The reward for processing j for (p_{j}) time units is (w_{j}p_{j}), and we are interested in constructing a feasible preemptive schedule such that the sum of rewards is maximized. We present a dynamic programming algorithm that solves the problem in pseudo-polynomial time and use it to obtain an FPTAS. Afterward, as our main contribution we propose an interesting efficient frontier approach for improved complexity bounds.

研究了处理时间连续可控的单机抢占调度中的加权吞吐量问题。可以在一台机器上调度一组任务。每个任务j都与一个非负权重(w_{j})、一个发布日期、一个到期日期和一个可能的处理时间间隔相关联。任务j可以使用给定间隔内的总处理时间(p_j)进行调度,也可以拒绝(不参与调度)。对于(p_{j})时间单位处理j的奖励是(w_{j}p_{j}),我们感兴趣的是构建一个可行的抢占式调度,使奖励总和最大化。提出了一种动态规划算法,该算法在伪多项式时间内解决了这一问题,并利用该算法得到了一个FPTAS。之后,作为我们的主要贡献,我们提出了一个有趣的有效的边界方法来改进复杂性边界。
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引用次数: 0
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Acta Informatica
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