Mark Cieliebak, S. Eidenbenz, Aris Pagourtzis, Konrad Schlude
The EQUAL SUM SUBSETS problem, where we are given a set of positive integers and we ask for two nonempty disjoint subsets such that their elements add up to the same total, is known to be NP-hard. In this paper we give (pseudo-)polynomial algorithms and/or (strong) NP-hardness proofs for several natural variations of EQUAL SUM SUBSETS. Among others we present (i) a framework for obtaining NP-hardness proofs and pseudopolynomial time algorithms for EQUAL SUM SUBSETS variations, which we apply to variants of the problem with additional selection restrictions, (ii) a proof of NP-hardness and a pseudo-polynomial time algorithm for the case where we ask for two subsets such that the ratio of their sums is some fixed rational r > 0, (iii) a pseudo-polynomial time algorithm for finding k subsets of equal sum, with k = O(1), and a proof of strong NP-hardness for the same problem with k = Ω(n), (iv) algorithms and hardness results for finding k equal sum subsets under the additional requirement that the subsets should be of equal cardinality. Our results are a step towards determining the dividing lines between polynomial time solvability, pseudo-polynomial time solvability, and strong NP-completeness of subset-sum related problems.
在EQUAL SUM SUBSETS问题中，我们给定一组正整数，并要求两个非空不相交的子集，使它们的元素加起来等于相同的总和，这是已知的NP-hard问题。本文给出了等和子集的几种自然变分的(伪)多项式算法和/或(强)np -硬度证明。等我们提出(i)的框架获得np困难的证明和pseudopolynomial总和相等时间算法子集变化,我们额外的选择适用于不同的问题限制,(2)一个np困难的证明和pseudo-polynomial时间算法的情况下我们要求两个子集,这样他们的资金是一些固定的比例合理r > 0, (iii) pseudo-polynomial时间算法寻找k总和相等的子集,与k = O (1),以及对于k = Ω的相同问题的强np -硬度的证明(n)， (iv)算法和在子集必须具有相等基数的附加要求下寻找k等和子集的硬度结果。我们的结果朝着确定子集和相关问题的多项式时间可解性、伪多项式时间可解性和强np完备性之间的分界线迈出了一步。
{"title":"On the Complexity of Variations of Equal Sum Subsets","authors":"Mark Cieliebak, S. Eidenbenz, Aris Pagourtzis, Konrad Schlude","doi":"10.5555/1737763.1737764","DOIUrl":"https://doi.org/10.5555/1737763.1737764","url":null,"abstract":"The EQUAL SUM SUBSETS problem, where we are given a set of positive integers and we ask for two nonempty disjoint subsets such that their elements add up to the same total, is known to be NP-hard. In this paper we give (pseudo-)polynomial algorithms and/or (strong) NP-hardness proofs for several natural variations of EQUAL SUM SUBSETS. Among others we present (i) a framework for obtaining NP-hardness proofs and pseudopolynomial time algorithms for EQUAL SUM SUBSETS variations, which we apply to variants of the problem with additional selection restrictions, (ii) a proof of NP-hardness and a pseudo-polynomial time algorithm for the case where we ask for two subsets such that the ratio of their sums is some fixed rational r > 0, (iii) a pseudo-polynomial time algorithm for finding k subsets of equal sum, with k = O(1), and a proof of strong NP-hardness for the same problem with k = Ω(n), (iv) algorithms and hardness results for finding k equal sum subsets under the additional requirement that the subsets should be of equal cardinality. \u0000 \u0000Our results are a step towards determining the dividing lines between polynomial time solvability, pseudo-polynomial time solvability, and strong NP-completeness of subset-sum related problems.","PeriodicalId":114503,"journal":{"name":"Nord. J. Comput.","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131598240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2006-06-01DOI: 10.7146/BRICS.V13I12.21917
Kristian Støvring
Higher-order matching is a special case of unification of simply-typed lambda-terms: in a matching equation, one of the two sides contains no unification variables. Loader has recently shown that higher-order matching up to beta equivalence is undecidable, but decidability of higher-order matching up to beta-eta equivalence is a long-standing open problem.We show that higher-order matching up to beta-eta equivalence is decidable if and only if a restricted form of higher-order matching up to beta equivalence is decidable: the restriction is that solutions must be in long beta-eta normal form.
{"title":"Higher-Order Beta Matching with Solutions in Long Beta-Eta Normal Form","authors":"Kristian Støvring","doi":"10.7146/BRICS.V13I12.21917","DOIUrl":"https://doi.org/10.7146/BRICS.V13I12.21917","url":null,"abstract":"Higher-order matching is a special case of unification of simply-typed lambda-terms: in a matching equation, one of the two sides contains no unification variables. Loader has recently shown that higher-order matching up to beta equivalence is undecidable, but decidability of higher-order matching up to beta-eta equivalence is a long-standing open problem.We show that higher-order matching up to beta-eta equivalence is decidable if and only if a restricted form of higher-order matching up to beta equivalence is decidable: the restriction is that solutions must be in long beta-eta normal form.","PeriodicalId":114503,"journal":{"name":"Nord. J. Comput.","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128556790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-04-01DOI: 10.13140/RG.2.2.36691.37920
Ragnhild Kobro Runde, Øystein Haugen, K. Stølen
STAIRS is an approach to the compositional development of UML interactions, such as sequence diagrams and interaction overview diagrams. An important aspect of STAIRS is the ability to distinguish between underspecification and inherent nondeterminism through the use of potential and mandatory alternatives. This paper investigates this distinction in more detail. Refinement notions explain when (and how) both kinds of nondeterminism may be reduced during the development process. In particular, in this paper we extend STAIRS with guards, which may be used to specify the choice between alternatives. Finally, we introduce the notion of an implementation and define what it means for an implementation to be correct with respect to a specification.
{"title":"Refining UML Interactions with Underspecification and Nondeterminism","authors":"Ragnhild Kobro Runde, Øystein Haugen, K. Stølen","doi":"10.13140/RG.2.2.36691.37920","DOIUrl":"https://doi.org/10.13140/RG.2.2.36691.37920","url":null,"abstract":"STAIRS is an approach to the compositional development of UML interactions, such as sequence diagrams and interaction overview diagrams. An important aspect of STAIRS is the ability to distinguish between underspecification and inherent nondeterminism through the use of potential and mandatory alternatives. This paper investigates this distinction in more detail. Refinement notions explain when (and how) both kinds of nondeterminism may be reduced during the development process. In particular, in this paper we extend STAIRS with guards, which may be used to specify the choice between alternatives. Finally, we introduce the notion of an implementation and define what it means for an implementation to be correct with respect to a specification.","PeriodicalId":114503,"journal":{"name":"Nord. J. Comput.","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127692992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Succinct Suffix Arrays based on Run-Length Encoding","authors":"V. Mäkinen, G. Navarro","doi":"10.1007/11496656_5","DOIUrl":"https://doi.org/10.1007/11496656_5","url":null,"abstract":"","PeriodicalId":114503,"journal":{"name":"Nord. J. Comput.","volume":"518 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123113799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-09-01DOI: 10.1007/3-540-36379-3_18
J. Fiala, P. Heggernes, Petter Kristiansen, J. A. Telle
{"title":"Generalized H-coloring and H-covering of Trees","authors":"J. Fiala, P. Heggernes, Petter Kristiansen, J. A. Telle","doi":"10.1007/3-540-36379-3_18","DOIUrl":"https://doi.org/10.1007/3-540-36379-3_18","url":null,"abstract":"","PeriodicalId":114503,"journal":{"name":"Nord. J. Comput.","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126862034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2003-06-30DOI: 10.1007/3-540-45061-0_89
S. Bereg, M. Segal
{"title":"Dynamic Algorithms for Approximating Interdistances","authors":"S. Bereg, M. Segal","doi":"10.1007/3-540-45061-0_89","DOIUrl":"https://doi.org/10.1007/3-540-45061-0_89","url":null,"abstract":"","PeriodicalId":114503,"journal":{"name":"Nord. J. Comput.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127373164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2002-03-01DOI: 10.1007/3-540-44693-1_44
G. Narasimhan, M. Smid
{"title":"Approximation Algorithms for the Bottleneck Stretch Factor Problem","authors":"G. Narasimhan, M. Smid","doi":"10.1007/3-540-44693-1_44","DOIUrl":"https://doi.org/10.1007/3-540-44693-1_44","url":null,"abstract":"","PeriodicalId":114503,"journal":{"name":"Nord. J. Comput.","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126323149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2002-03-01DOI: 10.7146/DPB.V31I556.7113
Erik Ernst
Multiple inheritance and similar mechanisms are usually only supported at compile time in statically typed languages. Nevertheless, dynamic multiple inheritance would be very useful in the development of complex systems, because it allows the creation of many related classes without an explosion in the size and level of redundancy in the source code. In fact, dynamic multiple inheritance is already available. The language gbeta is statically typed and has supported run-time combination of classes and methods since 1997, by means of the combination operator '&'. However, with certain combinations of operands the '&' operator fails; as a result, dynamic creation of new classes and methods was considered a dangerous operation in all cases. This paper presents a large and useful category of combinations, and proves that combinations in this category will always succeed.
{"title":"Safe Dynamic Multiple Inheritance","authors":"Erik Ernst","doi":"10.7146/DPB.V31I556.7113","DOIUrl":"https://doi.org/10.7146/DPB.V31I556.7113","url":null,"abstract":"Multiple inheritance and similar mechanisms are usually only supported at compile time in statically typed languages. Nevertheless, dynamic multiple inheritance would be very useful in the development of complex systems, because it allows the creation of many related classes without an explosion in the size and level of redundancy in the source code. In fact, dynamic multiple inheritance is already available. The language gbeta is statically typed and has supported run-time combination of classes and methods since 1997, by means of the combination operator '&'. However, with certain combinations of operands the '&' operator fails; as a result, dynamic creation of new classes and methods was considered a dangerous operation in all cases. This paper presents a large and useful category of combinations, and proves that combinations in this category will always succeed.","PeriodicalId":114503,"journal":{"name":"Nord. J. Comput.","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2002-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130354139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2000-07-05DOI: 10.1007/3-540-44985-X_29
P. Agarwal, L. Guibas, Sariel Har-Peled, Alexander Rabinovitch, M. Sharir
{"title":"Penetration Depth of Two Convex Polytopes in 3D","authors":"P. Agarwal, L. Guibas, Sariel Har-Peled, Alexander Rabinovitch, M. Sharir","doi":"10.1007/3-540-44985-X_29","DOIUrl":"https://doi.org/10.1007/3-540-44985-X_29","url":null,"abstract":"","PeriodicalId":114503,"journal":{"name":"Nord. J. Comput.","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132910395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2000-06-01DOI: 10.7146/BRICS.V7I31.20165
Bernd Grobauer, J. Lawall
Specialization of a string matcher is a canonical example of partial evaluation. A naive implementation of a string matcher repeatedly matches a pattern against every substring of the data string; this operation should intuitively benefit from specializing the matcher with respect to the pattern. In practice, however, producing an efficient implementation by performing this specialization using standard partial-evaluation techniques requires non-trivial binding-time improvements. Starting with a naive matcher, we thus present a derivation of such a binding-time improved string matcher. We show that specialization with respect to a pattern yields a matcher with code size linear in the length of the pattern and a running time independent of the length of the pattern and linear in the length of the data string. We then consider several variants of matchers that specialize well, amongst them the first such matcher presented in the literature, and we demonstrate how variants can be derived from each other systematically.
{"title":"Partial Evaluation of Pattern Matching in Strings, revisited","authors":"Bernd Grobauer, J. Lawall","doi":"10.7146/BRICS.V7I31.20165","DOIUrl":"https://doi.org/10.7146/BRICS.V7I31.20165","url":null,"abstract":"Specialization of a string matcher is a canonical example of partial evaluation. A naive implementation of a string matcher repeatedly matches a pattern against every substring of the data string; this operation should intuitively benefit from specializing the matcher with respect to the pattern. In practice, however, producing an efficient implementation by performing this specialization using standard partial-evaluation techniques requires non-trivial binding-time improvements. Starting with a naive matcher, we thus present a derivation of such a binding-time improved string matcher. We show that specialization with respect to a pattern yields a matcher with code size linear in the length of the pattern and a running time independent of the length of the pattern and linear in the length of the data string. We then consider several variants of matchers that specialize well, amongst them the first such matcher presented in the literature, and we demonstrate how variants can be derived from each other systematically.","PeriodicalId":114503,"journal":{"name":"Nord. J. Comput.","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122865261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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