Pub Date : 2022-08-04DOI: 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).
{"title":"On the decidability of finding a positive ILP-instance in a regular set of ILP-instances","authors":"Petra Wolf","doi":"10.1007/s00236-022-00429-x","DOIUrl":"10.1007/s00236-022-00429-x","url":null,"abstract":"<div><p>The regular intersection emptiness problem for a decision problem <i>P</i> (<span>({{textit{int}}_{{mathrm {Reg}}}})</span>(<i>P</i>)) is to decide whether a potentially infinite regular set of encoded <i>P</i>-instances contains a positive one. Since <span>({{textit{int}}_{{mathrm {Reg}}}})</span>(<i>P</i>) 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 <span>({{textit{int}}_{{mathrm {Reg}}}})</span>-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 <span>({{textit{int}}_{{mathrm {Reg}}}})</span>-problem for the well-known NP-complete problem <span>Integer Linear Programming</span> (<span>ILP</span>). It is shown that any DFA that describes a set of <span>ILP</span>-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 <span>({{textit{int}}_{{mathrm {Reg}}}})</span>(<span>ILP</span>).</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":"59 4","pages":"505 - 519"},"PeriodicalIF":0.6,"publicationDate":"2022-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00236-022-00429-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41604344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-27DOI: 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.
{"title":"Pushdown automata and constant height: decidability and bounds","authors":"Giovanni Pighizzini, Luca Prigioniero","doi":"10.1007/s00236-022-00434-0","DOIUrl":"10.1007/s00236-022-00434-0","url":null,"abstract":"<div><p>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.\u0000</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":"60 2","pages":"123 - 144"},"PeriodicalIF":0.6,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00236-022-00434-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42136940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-25DOI: 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.
{"title":"Reoptimization of parameterized problems","authors":"Hans-Joachim Böckenhauer, Elisabet Burjons, Martin Raszyk, Peter Rossmanith","doi":"10.1007/s00236-022-00428-y","DOIUrl":"10.1007/s00236-022-00428-y","url":null,"abstract":"<div><p>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 <span>(mathbf{NP}subseteq mathbf{coNP/poly})</span>. 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 <span>Connected Vertex Cover</span> does not have a polynomial kernel unless <span>Set Cover</span> has a polynomial compression. In a different direction, looking at problems with polynomial kernels, we find that the reoptimization version of <span>Vertex Cover</span> has a polynomial kernel of size <span>(varvec{2k+1})</span> using crown decompositions only, which improves the size of the kernel achievable with this technique in the classic problem.</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":"59 4","pages":"427 - 450"},"PeriodicalIF":0.6,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00236-022-00428-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"40334166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-25DOI: 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.
{"title":"The regular languages of wire linear AC(^0)","authors":"Michaël Cadilhac, Charles Paperman","doi":"10.1007/s00236-022-00432-2","DOIUrl":"10.1007/s00236-022-00432-2","url":null,"abstract":"<div><p>In this paper, the regular languages of wire linear <span>(hbox {AC}^0)</span>are characterized as the languages expressible in the two-variable fragment of first-order logic with regular predicates, <span>(mathrm{FO}^2[mathrm{reg}])</span>. Additionally, they are characterized as the languages recognized by the algebraic class <span>(mathbf {QLDA})</span>. The class is shown to be decidable and examples of languages in and outside of it are presented.</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":"59 4","pages":"321 - 336"},"PeriodicalIF":0.6,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50102664","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 : 2022-07-22DOI: 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.
{"title":"Careful synchronization of partial deterministic finite automata","authors":"Hanan Shabana, M. V. Volkov","doi":"10.1007/s00236-022-00433-1","DOIUrl":"10.1007/s00236-022-00433-1","url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":"59 4","pages":"479 - 504"},"PeriodicalIF":0.6,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50042538","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 : 2022-07-20DOI: 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。
{"title":"Operational complexity and pumping lemmas","authors":"Jürgen Dassow, Ismaël Jecker","doi":"10.1007/s00236-022-00431-3","DOIUrl":"10.1007/s00236-022-00431-3","url":null,"abstract":"<div><p>The well-known pumping lemma for regular languages states that, for any regular language <i>L</i>, there is a constant <i>p</i> (depending on <i>L</i>) such that the following holds: If <span>(win L)</span> and <span>(vert wvert ge p)</span>, then there are words <span>(xin V^{*})</span>, <span>(yin V^+)</span>, and <span>(zin V^{*})</span> such that <span>(w=xyz)</span> and <span>(xy^tzin L)</span> for <span>(tge 0)</span>. The minimal pumping constant <span>({{{,mathrm{mpc},}}(L)})</span> of <i>L</i> is the minimal number <i>p</i> for which the conditions of the pumping lemma are satisfied. We investigate the behaviour of <span>({{{,mathrm{mpc},}}})</span> with respect to operations, i. e., for an <i>n</i>-ary regularity preserving operation <span>(circ )</span>, we study the set <span>({g_{circ }^{{{,mathrm{mpc},}}}(k_1,k_2,ldots ,k_n)})</span> of all numbers <i>k</i> such that there are regular languages <span>(L_1,L_2,ldots ,L_n)</span> with <span>({{{,mathrm{mpc},}}(L_i)=k_i})</span> for <span>(1le ile n)</span> and <span>({{{,mathrm{mpc},}}(circ (L_1,L_2,ldots ,L_n)=~k})</span>. With respect to Kleene closure, complement, reversal, prefix and suffix-closure, circular shift, union, intersection, set-subtraction, symmetric difference,and concatenation, we determine <span>({g_{circ }^{{{,mathrm{mpc},}}}(k_1,k_2,ldots ,k_n)})</span> completely. Furthermore, we give some results with respect to the minimal pumping length where, in addition, <span>(vert xyvert le p)</span> has to hold.</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":"59 4","pages":"337 - 355"},"PeriodicalIF":0.6,"publicationDate":"2022-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00236-022-00431-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43609320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-19DOI: 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.
{"title":"Weighted throughput in a single machine preemptive scheduling with continuous controllable processing times","authors":"Asaf Levin, Tal Shusterman","doi":"10.1007/s00236-022-00430-4","DOIUrl":"10.1007/s00236-022-00430-4","url":null,"abstract":"<div><p>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 <i>j</i> is associated with a nonnegative weight <span>(w_{j})</span>, a release date, a due date, and an interval of possible processing times. A task <i>j</i> can either be scheduled with a total processing time <span>(p_j)</span> which is in the given interval, or rejected (not participating in the schedule). The reward for processing <i>j</i> for <span>(p_{j})</span> time units is <span>(w_{j}p_{j})</span>, 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.</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":"60 2","pages":"101 - 122"},"PeriodicalIF":0.6,"publicationDate":"2022-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44782247","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 : 2022-06-26DOI: 10.1007/s00236-022-00424-2
Dietrich Kuske, Christian Schwarz
This paper considers the structure consisting of the set of all words over a given alphabet together with the subword relation, regular predicates, and constants for every word. We are interested in the counting extension of first-order logic by threshold counting quantifiers. The main result shows that the two-variable fragment of this logic can be decided in twofold exponential alternating time with linearly many alternations (and therefore in particular in twofold exponential space as announced in the conference version (Kuske and Schwarz, in: MFCS’20, Leibniz International Proceedings in Informatics (LIPIcs) vol. 170, pp 56:1–56:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020) of this paper) provided the regular predicates are restricted to piecewise testable ones. This result improves prior insights by Karandikar and Schnoebelen by extending the logic and saving one exponent in the space bound. Its proof consists of two main parts: First, we provide a quantifier elimination procedure that results in a formula with constants of bounded length (this generalises the procedure by Karandikar and Schnoebelen for first-order logic). From this, it follows that quantification in formulas can be restricted to words of bounded length, i.e., the second part of the proof is an adaptation of the method by Ferrante and Rackoff to counting logic and deviates significantly from the path of reasoning by Karandikar and Schnoebelen.
{"title":"Alternating complexity of counting first-order logic for the subword order","authors":"Dietrich Kuske, Christian Schwarz","doi":"10.1007/s00236-022-00424-2","DOIUrl":"10.1007/s00236-022-00424-2","url":null,"abstract":"<div><p>This paper considers the structure consisting of the set of all words over a given alphabet together with the subword relation, regular predicates, and constants for every word. We are interested in the counting extension of first-order logic by threshold counting quantifiers. The main result shows that the two-variable fragment of this logic can be decided in twofold exponential alternating time with linearly many alternations (and therefore in particular in twofold exponential space as announced in the conference version (Kuske and Schwarz, in: MFCS’20, Leibniz International Proceedings in Informatics (LIPIcs) vol. 170, pp 56:1–56:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020) of this paper) provided the regular predicates are restricted to piecewise testable ones. This result improves prior insights by Karandikar and Schnoebelen by extending the logic and saving one exponent in the space bound. Its proof consists of two main parts: First, we provide a quantifier elimination procedure that results in a formula with constants of bounded length (this generalises the procedure by Karandikar and Schnoebelen for first-order logic). From this, it follows that quantification in formulas can be restricted to words of bounded length, i.e., the second part of the proof is an adaptation of the method by Ferrante and Rackoff to counting logic and deviates significantly from the path of reasoning by Karandikar and Schnoebelen.</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":"60 1","pages":"79 - 100"},"PeriodicalIF":0.6,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00236-022-00424-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42991457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}