Pub Date : 2023-12-11DOI: 10.1007/s00224-023-10154-8
Joel Day, Vijay Ganesh, Nathan Grewal, Matthew Konefal, Florin Manea
Word equations are equations (alpha doteq beta ) where (alpha ) and (beta ) are words consisting of letters from some alphabet (Sigma ) and variables from a set X. Recently, there has been substantial interest in the context of string solving in logics combining word equations with other kinds of constraints on words such as (regular) language membership (regular constraints) and arithmetic over string lengths (length constraints). We consider the expressive power of such logics by looking at the set of all values a single variable might take as part of a satisfying assignment for a given formula. Hence, each formula-variable pair defines a formal language, and each logic defines a class of formal languages. We consider logics arising from combining word equations with either length constraints, regular constraints, or both. We also consider word equations with visibly pushdown language membership constraints as a generalisation of the combination of regular and length constraints. We show that word equations with visibly pushdown membership constraints are sufficient to express all recursively enumerable languages and hence satisfiability is undecidable in this case. We then establish a strict hierarchy involving the other combinations. We also provide a complete characterisation of when a thin regular language is expressible by word equations (alone) and some further partial results for regular languages in the general case.
{"title":"A Closer Look at the Expressive Power of Logics Based on Word Equations","authors":"Joel Day, Vijay Ganesh, Nathan Grewal, Matthew Konefal, Florin Manea","doi":"10.1007/s00224-023-10154-8","DOIUrl":"https://doi.org/10.1007/s00224-023-10154-8","url":null,"abstract":"<p>Word equations are equations <span>(alpha doteq beta )</span> where <span>(alpha )</span> and <span>(beta )</span> are words consisting of letters from some alphabet <span>(Sigma )</span> and variables from a set <i>X</i>. Recently, there has been substantial interest in the context of string solving in logics combining word equations with other kinds of constraints on words such as (regular) language membership (regular constraints) and arithmetic over string lengths (length constraints). We consider the expressive power of such logics by looking at the set of all values a single variable might take as part of a satisfying assignment for a given formula. Hence, each formula-variable pair defines a formal language, and each logic defines a class of formal languages. We consider logics arising from combining word equations with either length constraints, regular constraints, or both. We also consider word equations with visibly pushdown language membership constraints as a generalisation of the combination of regular and length constraints. We show that word equations with visibly pushdown membership constraints are sufficient to express all recursively enumerable languages and hence satisfiability is undecidable in this case. We then establish a strict hierarchy involving the other combinations. We also provide a complete characterisation of when a thin regular language is expressible by word equations (alone) and some further partial results for regular languages in the general case.</p>","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138566860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-04DOI: 10.1007/s00224-023-10153-9
Pierre Bergé, Guillaume Ducoffe, Michel Habib
On sparse graphs, Roditty and Williams [2013] proved that no (varvec{O(n^{2-varepsilon })})-time algorithm achieves an approximation factor smaller than (varvec{frac{3}{2}}) for the diameter problem unless SETH fails. In this article, we solve an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier? We propose the first combinatorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for all eccentricities in median graphs with bounded dimension (varvec{d}), i.e. the dimension of the largest induced hypercube. This prerequisite on (varvec{d}) is not necessary anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is (varvec{O(n^{1.6456}log ^{O(1)} n)}). We provide also some satellite outcomes related to this general result. In particular, restricted to simplex graphs, this algorithm enumerates all eccentricities with a quasilinear running time. Moreover, an algorithm is proposed to compute exactly all reach centralities in time (varvec{O(2^{3d}nlog ^{O(1)}n)}).
{"title":"Subquadratic-time Algorithm for the Diameter and all Eccentricities on Median Graphs","authors":"Pierre Bergé, Guillaume Ducoffe, Michel Habib","doi":"10.1007/s00224-023-10153-9","DOIUrl":"https://doi.org/10.1007/s00224-023-10153-9","url":null,"abstract":"<p>On sparse graphs, Roditty and Williams [2013] proved that no <span>(varvec{O(n^{2-varepsilon })})</span>-time algorithm achieves an approximation factor smaller than <span>(varvec{frac{3}{2}})</span> for the diameter problem unless SETH fails. In this article, we solve an open question formulated in the literature: can we use the structural properties of median graphs to break this global quadratic barrier? We propose the first combinatorial algorithm computing exactly all eccentricities of a median graph in truly subquadratic time. Median graphs constitute the family of graphs which is the most studied in metric graph theory because their structure represents many other discrete and geometric concepts, such as CAT(0) cube complexes. Our result generalizes a recent one, stating that there is a linear-time algorithm for all eccentricities in median graphs with bounded dimension <span>(varvec{d})</span>, <i>i.e.</i> the dimension of the largest induced hypercube. This prerequisite on <span>(varvec{d})</span> is not necessary anymore to determine all eccentricities in subquadratic time. The execution time of our algorithm is <span>(varvec{O(n^{1.6456}log ^{O(1)} n)})</span>. We provide also some satellite outcomes related to this general result. In particular, restricted to simplex graphs, this algorithm enumerates all eccentricities with a quasilinear running time. Moreover, an algorithm is proposed to compute exactly all reach centralities in time <span>(varvec{O(2^{3d}nlog ^{O(1)}n)})</span>.</p>","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138542830","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.1007/s00224-023-10152-w
Vahan Mkrtchyan, Garik Petrosyan, K. Subramani, Piotr Wojciechowski
In this paper, we examine variants of the partial vertex cover problem from the perspective of parameterized algorithms. Recall that in the classical vertex cover problem (VC), we are given a graph (mathbf{G = langle V, E rangle }) and a number k and asked if we can cover all of the edges in (textbf{E}), using at most k vertices from (textbf{V}). The partial vertex cover problem (PVC) is a more general version of the VC problem in which we are given an additional parameter (k'). We then ask the question of whether at least (k') of the edges in (textbf{E}) can be covered using at most k vertices from (textbf{V}). Note that the VC problem is a special case of the PVC problem when (k'=|textbf{E}|). In this paper, we study the weighted generalizations of the PVC problem. This is called the weighted partial vertex cover problem (WPVC). In the WPVC problem, we are given two parameters R and L, associated respectively with the vertex set (textbf{V}) and edge set (textbf{E}) of the graph (textbf{G}) respectively. Additionally, we are given non-negative integral weight functions for the vertices and the edges. The goal then is to cover edges of total weight at least L, using vertices of total weight at most R. This paper studies several variants of the PVC and WPVC problems and establishes new results from the perspective of fixed-parameter tractability and W[1]-hardness. We also introduce a new problem called the partial vertex cover with matching constraints and show that it is Fixed-Parameter Tractable (FPT) for a certain class of graphs. Finally, we show that the WPVC problem is APX-complete for bipartite graphs.
本文从参数化算法的角度研究了部分顶点覆盖问题的变体。回想一下,在经典的顶点覆盖问题(VC)中,我们给定一个图(mathbf{G = langle V, E rangle })和一个数字k,并问我们是否可以覆盖(textbf{E})中的所有边,使用(textbf{V})中的最多k个顶点。部分顶点覆盖问题(PVC)是VC问题的一个更一般的版本,在这个版本中,我们得到了一个额外的参数(k')。然后我们问,是否(textbf{E})中至少(k')条边可以使用(textbf{V})中最多k个顶点来覆盖。注意,VC问题是PVC问题的特例当(k'=|textbf{E}|)。本文研究了PVC问题的加权推广。这被称为加权部分顶点覆盖问题(WPVC)。在WPVC问题中,我们给出两个参数R和L,分别与图(textbf{G})的顶点集(textbf{V})和边集(textbf{E})相关联。此外,我们给出了顶点和边的非负积分权函数。然后,目标是覆盖总权值至少为L的边,使用总权值最多为r的顶点。本文研究了PVC和WPVC问题的几种变体,并从固定参数可追溯性和W[1]-硬度的角度建立了新的结果。我们还引入了具有匹配约束的部分顶点覆盖问题,并证明了它对于某一类图是固定参数可处理的(FPT)。最后,我们证明了WPVC问题对于二部图是apx完全的。
{"title":"On the Partial Vertex Cover Problem in Bipartite Graphs - a Parameterized Perspective","authors":"Vahan Mkrtchyan, Garik Petrosyan, K. Subramani, Piotr Wojciechowski","doi":"10.1007/s00224-023-10152-w","DOIUrl":"https://doi.org/10.1007/s00224-023-10152-w","url":null,"abstract":"<p>In this paper, we examine variants of the partial vertex cover problem from the perspective of parameterized algorithms. Recall that in the classical vertex cover problem (VC), we are given a graph <span>(mathbf{G = langle V, E rangle })</span> and a number <i>k</i> and asked if we can cover all of the edges in <span>(textbf{E})</span>, using at most <i>k</i> vertices from <span>(textbf{V})</span>. The partial vertex cover problem (PVC) is a more general version of the VC problem in which we are given an additional parameter <span>(k')</span>. We then ask the question of whether at least <span>(k')</span> of the edges in <span>(textbf{E})</span> can be covered using at most <i>k</i> vertices from <span>(textbf{V})</span>. Note that the VC problem is a special case of the PVC problem when <span>(k'=|textbf{E}|)</span>. In this paper, we study the weighted generalizations of the PVC problem. This is called the weighted partial vertex cover problem (WPVC). In the WPVC problem, we are given two parameters <i>R</i> and <i>L</i>, associated respectively with the vertex set <span>(textbf{V})</span> and edge set <span>(textbf{E})</span> of the graph <span>(textbf{G})</span> respectively. Additionally, we are given non-negative integral weight functions for the vertices and the edges. The goal then is to cover edges of total weight at least <i>L</i>, using vertices of total weight at most <i>R</i>. This paper studies several variants of the PVC and WPVC problems and establishes new results from the perspective of fixed-parameter tractability and <b>W[1]-hardness</b>. We also introduce a new problem called the partial vertex cover with matching constraints and show that it is Fixed-Parameter Tractable (<b>FPT</b>) for a certain class of graphs. Finally, we show that the WPVC problem is <b>APX-complete</b> for bipartite graphs.</p>","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138524057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-14DOI: 10.1007/s00224-023-10150-y
Vincent Froese, Pascal Kunz, Philipp Zschoche
Abstract We study the network untangling problem introduced by Rozenshtein et al. (Data Min. Knowl. Disc. 35(1), 213–247, 2021), which is a variant of Vertex Cover on temporal graphs–graphs whose edge set changes over discrete time steps. They introduce two problem variants. The goal is to select at most k time intervals for each vertex such that all time-edges are covered and (depending on the problem variant) either the maximum interval length or the total sum of interval lengths is minimized. This problem has data mining applications in finding activity timelines that explain the interactions of entities in complex networks. Both variants of the problem are NP-hard. In this paper, we initiate a multivariate complexity analysis involving the following parameters: number of vertices, lifetime of the temporal graph, number of intervals per vertex, and the interval length bound. For both problem versions, we (almost) completely settle the parameterized complexity for all combinations of those four parameters, thereby delineating the border of fixed-parameter tractability.
摘要本文研究了Rozenshtein等人提出的网络解缠问题(Data Min. knowledge)。Disc. 35(1), 213-247, 2021),它是时间图(其边缘集在离散时间步长上变化的图)上的顶点覆盖的一种变体。它们引入了两个问题变体。目标是为每个顶点选择最多k个时间间隔,以便覆盖所有时间边,并且(取决于问题的变体)最小化最大间隔长度或间隔长度的总和。这个问题有数据挖掘应用在寻找解释复杂网络中实体相互作用的活动时间轴上。这个问题的两个变体都是np困难的。在本文中,我们开始了一个多元复杂性分析,涉及以下参数:顶点数,时间图的生存期,每个顶点的区间数和区间长度界。对于这两个问题版本,我们(几乎)完全解决了这四个参数的所有组合的参数化复杂性,从而划定了固定参数可跟踪性的边界。
{"title":"Disentangling the Computational Complexity of Network Untangling","authors":"Vincent Froese, Pascal Kunz, Philipp Zschoche","doi":"10.1007/s00224-023-10150-y","DOIUrl":"https://doi.org/10.1007/s00224-023-10150-y","url":null,"abstract":"Abstract We study the network untangling problem introduced by Rozenshtein et al. (Data Min. Knowl. Disc. 35(1), 213–247, 2021), which is a variant of Vertex Cover on temporal graphs–graphs whose edge set changes over discrete time steps. They introduce two problem variants. The goal is to select at most k time intervals for each vertex such that all time-edges are covered and (depending on the problem variant) either the maximum interval length or the total sum of interval lengths is minimized. This problem has data mining applications in finding activity timelines that explain the interactions of entities in complex networks. Both variants of the problem are NP-hard. In this paper, we initiate a multivariate complexity analysis involving the following parameters: number of vertices, lifetime of the temporal graph, number of intervals per vertex, and the interval length bound. For both problem versions, we (almost) completely settle the parameterized complexity for all combinations of those four parameters, thereby delineating the border of fixed-parameter tractability.","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134992133","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-07DOI: 10.1007/s00224-023-10149-5
Frank Gurski, Jörg Rothe, Robin Weishaupt
Abstract Frei et al. (J. Comput. Syst. Sci. 123 , 103–121, 2022) show that the stability, vertex stability, and unfrozenness problems with respect to certain graph parameters are complete for $$varvec{Theta _{2}^{textrm{P}}}$$ Θ2P , the class of problems solvable in polynomial time by parallel access to an NP oracle. They studied the common graph parameters $$varvec{alpha }$$ α (the independence number), $$varvec{beta }$$ β (the vertex cover number), $$varvec{omega }$$ ω (the clique number), and $$varvec{chi }$$ χ (the chromatic number). We complement their approach by providing polynomial-time algorithms solving these problems for special graph classes, namely for graphs with bounded tree-width or bounded clique-width. In order to improve these general time bounds even further, we then focus on trees, forests, bipartite graphs, and co-graphs.
{"title":"Stability, Vertex Stability, and Unfrozenness for Special Graph Classes","authors":"Frank Gurski, Jörg Rothe, Robin Weishaupt","doi":"10.1007/s00224-023-10149-5","DOIUrl":"https://doi.org/10.1007/s00224-023-10149-5","url":null,"abstract":"Abstract Frei et al. (J. Comput. Syst. Sci. 123 , 103–121, 2022) show that the stability, vertex stability, and unfrozenness problems with respect to certain graph parameters are complete for $$varvec{Theta _{2}^{textrm{P}}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msubsup> <mml:mi>Θ</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mtext>P</mml:mtext> </mml:msubsup> </mml:mrow> </mml:math> , the class of problems solvable in polynomial time by parallel access to an NP oracle. They studied the common graph parameters $$varvec{alpha }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> </mml:math> (the independence number), $$varvec{beta }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> </mml:math> (the vertex cover number), $$varvec{omega }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> (the clique number), and $$varvec{chi }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>χ</mml:mi> </mml:mrow> </mml:math> (the chromatic number). We complement their approach by providing polynomial-time algorithms solving these problems for special graph classes, namely for graphs with bounded tree-width or bounded clique-width. In order to improve these general time bounds even further, we then focus on trees, forests, bipartite graphs, and co-graphs.","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135476611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-24DOI: 10.1007/s00224-023-10146-8
Qisheng Wang, Mingsheng Ying
Abstract Lexicographically minimal string rotation (LMSR) is a problem to find the minimal one among all rotations of a string in the lexicographical order, which is widely used in equality checking of graphs, polygons, automata and chemical structures. In this paper, we propose an $$O(n^{3/4})$$ O(n3/4) quantum query algorithm for LMSR. In particular, the algorithm has average-case query complexity $$O(sqrt{n} log n)$$ O(nlogn) , which is shown to be asymptotically optimal up to a polylogarithmic factor, compared to its $$Omega left( sqrt{n/log n}right) $$ Ωn/logn lower bound. Furthermore, we show that our quantum algorithm outperforms any (classical) randomized algorithms in both worst and average cases. As an application, it is used in benzenoid identification and disjoint-cycle automata minimization.
字典最小字符串旋转(LMSR)是一个在字典顺序的所有字符串旋转中找到最小值的问题,广泛应用于图、多边形、自动机和化学结构的相等性检验。本文提出了一种用于LMSR的$$O(n^{3/4})$$ O (n 3 / 4)量子查询算法。特别是,该算法具有平均情况下的查询复杂度$$O(sqrt{n} log n)$$ O (n log n),与其$$Omega left( sqrt{n/log n}right) $$ Ω n / log n下界相比,它被证明是渐近最优的,直到一个多对数因子。此外,我们表明我们的量子算法在最差和平均情况下都优于任何(经典)随机化算法。作为一个应用,它被用于苯类识别和分离循环自动机最小化。
{"title":"Quantum Algorithm for Lexicographically Minimal String Rotation","authors":"Qisheng Wang, Mingsheng Ying","doi":"10.1007/s00224-023-10146-8","DOIUrl":"https://doi.org/10.1007/s00224-023-10146-8","url":null,"abstract":"Abstract Lexicographically minimal string rotation (LMSR) is a problem to find the minimal one among all rotations of a string in the lexicographical order, which is widely used in equality checking of graphs, polygons, automata and chemical structures. In this paper, we propose an $$O(n^{3/4})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> quantum query algorithm for LMSR. In particular, the algorithm has average-case query complexity $$O(sqrt{n} log n)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , which is shown to be asymptotically optimal up to a polylogarithmic factor, compared to its $$Omega left( sqrt{n/log n}right) $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mfenced> <mml:msqrt> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>/</mml:mo> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msqrt> </mml:mfenced> </mml:mrow> </mml:math> lower bound. Furthermore, we show that our quantum algorithm outperforms any (classical) randomized algorithms in both worst and average cases. As an application, it is used in benzenoid identification and disjoint-cycle automata minimization.","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135273543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-19DOI: 10.1007/s00224-023-10147-7
David E. Brown, David Skidmore
{"title":"Representing the Integer Factorization Problem Using Ordered Binary Decision Diagrams","authors":"David E. Brown, David Skidmore","doi":"10.1007/s00224-023-10147-7","DOIUrl":"https://doi.org/10.1007/s00224-023-10147-7","url":null,"abstract":"","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135667772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-06DOI: 10.1007/s00224-023-10145-9
Abdolhamid Ghodselahi, Fabian Kuhn
Abstract We introduce an online variant of mobile facility location (MFL) (introduced by Demaine et al. (SODA 258–267 2007)). We call this new problem online mobile facility location (OMFL). In the OMFL problem, initially, we are given a set of k mobile facilities with their starting locations. One by one, requests are added. After each request arrives, one can make some changes to the facility locations before the subsequent request arrives. Each request is always assigned to the nearest facility. The cost of this assignment is the distance from the request to the facility. The objective is to minimize the total cost, which consists of the relocation cost of facilities and the distance cost of requests to their nearest facilities. We provide a lower bound for the OMFL problem that even holds on uniform metrics . A natural approach to solve the OMFL problem for general metric spaces is to utilize hierarchically well-separated trees (HSTs) and directly solve the OMFL problem on HSTs. In this paper, we provide the first step in this direction by solving a generalized variant of the OMFL problem on uniform metrics that we call G-OMFL. We devise a simple deterministic online algorithm and provide a tight analysis for the algorithm. The second step remains an open question. Inspired by the k -server problem, we introduce a new variant of the OMFL problem that focuses solely on minimizing movement cost. We refer to this variant as M-OMFL. Additionally, we provide a lower bound for M-OMFL that is applicable even on uniform metrics.
{"title":"Toward Online Mobile Facility Location on General Metrics","authors":"Abdolhamid Ghodselahi, Fabian Kuhn","doi":"10.1007/s00224-023-10145-9","DOIUrl":"https://doi.org/10.1007/s00224-023-10145-9","url":null,"abstract":"Abstract We introduce an online variant of mobile facility location (MFL) (introduced by Demaine et al. (SODA 258–267 2007)). We call this new problem online mobile facility location (OMFL). In the OMFL problem, initially, we are given a set of k mobile facilities with their starting locations. One by one, requests are added. After each request arrives, one can make some changes to the facility locations before the subsequent request arrives. Each request is always assigned to the nearest facility. The cost of this assignment is the distance from the request to the facility. The objective is to minimize the total cost, which consists of the relocation cost of facilities and the distance cost of requests to their nearest facilities. We provide a lower bound for the OMFL problem that even holds on uniform metrics . A natural approach to solve the OMFL problem for general metric spaces is to utilize hierarchically well-separated trees (HSTs) and directly solve the OMFL problem on HSTs. In this paper, we provide the first step in this direction by solving a generalized variant of the OMFL problem on uniform metrics that we call G-OMFL. We devise a simple deterministic online algorithm and provide a tight analysis for the algorithm. The second step remains an open question. Inspired by the k -server problem, we introduce a new variant of the OMFL problem that focuses solely on minimizing movement cost. We refer to this variant as M-OMFL. Additionally, we provide a lower bound for M-OMFL that is applicable even on uniform metrics.","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135350992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-03DOI: 10.1007/s00224-023-10144-w
Andreas Maletti, Andreea-Teodora Nász
Abstract The HOM problem, which asks whether the image of a regular tree language under a given tree homomorphism is again regular, is known to be decidable [Godoy & Giménez: The HOM problem is decidable. JACM 60(4), 2013]. However, the problem remains open for regular weighted tree languages. It is demonstrated that the main notion used in the unweighted setting, the tree automaton with equality and inequality constraints , can straightforwardly be generalized to the weighted setting and can represent the image of any regular weighted tree language under any nondeleting and nonerasing tree homomorphism. Several closure properties as well as decision problems are also investigated for the weighted tree languages generated by weighted tree automata with constraints.
{"title":"Weighted Tree Automata with Constraints","authors":"Andreas Maletti, Andreea-Teodora Nász","doi":"10.1007/s00224-023-10144-w","DOIUrl":"https://doi.org/10.1007/s00224-023-10144-w","url":null,"abstract":"Abstract The HOM problem, which asks whether the image of a regular tree language under a given tree homomorphism is again regular, is known to be decidable [Godoy & Giménez: The HOM problem is decidable. JACM 60(4), 2013]. However, the problem remains open for regular weighted tree languages. It is demonstrated that the main notion used in the unweighted setting, the tree automaton with equality and inequality constraints , can straightforwardly be generalized to the weighted setting and can represent the image of any regular weighted tree language under any nondeleting and nonerasing tree homomorphism. Several closure properties as well as decision problems are also investigated for the weighted tree languages generated by weighted tree automata with constraints.","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135697010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-23DOI: 10.1007/s00224-023-10132-0
Lars Jaffke, Paloma T. Lima, Daniel Lokshtanov
Abstract We provide a polynomial-time algorithm for b -Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial time results on graph classes, and answers open questions posed by Campos and Silva (Algorithmica 80 (1), 104–115, 2018) and Bonomo et al. (Graphs and Combinatorics 25 (2), 153–167, 2009). This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is $$textsf{FPT}$$ FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound. The running times of the clique-width based algorithms for $$b$$ b - Coloring and Fall Coloring are tight under the Exponential Time Hypothesis.
摘要给出了常团宽图上b -上色的多项式时间算法。这统一并扩展了几乎所有已知的图类多项式时间结果,并回答了Campos和Silva (Algorithmica 80(1), 104-115, 2018)和Bonomo等人(Graphs and Combinatorics 25(2), 153-167, 2009)提出的开放问题。这是关于这个问题的结构参数化的第一个结果。我们证明了在一般图上用顶点覆盖数参数化的问题是$$textsf{FPT}$$ FPT,在弦图上用颜色数参数化的问题是 FPT。此外,我们观察到我们的有界团宽度图的算法可以在相同的运行时间范围内适用于解决Fall Coloring问题。在指数时间假设下,基于团宽度的$$b$$ b -着色和Fall着色算法的运行时间较紧。
{"title":"b-Coloring Parameterized by Clique-Width","authors":"Lars Jaffke, Paloma T. Lima, Daniel Lokshtanov","doi":"10.1007/s00224-023-10132-0","DOIUrl":"https://doi.org/10.1007/s00224-023-10132-0","url":null,"abstract":"Abstract We provide a polynomial-time algorithm for b -Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial time results on graph classes, and answers open questions posed by Campos and Silva (Algorithmica 80 (1), 104–115, 2018) and Bonomo et al. (Graphs and Combinatorics 25 (2), 153–167, 2009). This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is $$textsf{FPT}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>FPT</mml:mi> </mml:math> when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound. The running times of the clique-width based algorithms for $$b$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>b</mml:mi> </mml:math> - Coloring and Fall Coloring are tight under the Exponential Time Hypothesis.","PeriodicalId":22832,"journal":{"name":"Theory of Computing Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135959829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}