Pub Date : 2023-09-01DOI: 10.1007/s10601-023-09355-2
Kyle E. C. Booth
{"title":"Constraint programming approaches to electric vehicle and robot routing problems","authors":"Kyle E. C. Booth","doi":"10.1007/s10601-023-09355-2","DOIUrl":"https://doi.org/10.1007/s10601-023-09355-2","url":null,"abstract":"","PeriodicalId":127439,"journal":{"name":"Constraints - An International Journal","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135389871","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 : 2023-09-01DOI: 10.1007/s10601-023-09360-5
Christian Bessiere, Clément Carbonnel, Martin C. Cooper, Emmanuel Hebrard
Explaining the outcome of programs has become one of the main concerns in AI research. In constraint programming, a user may want the system to explain why a given variable assignment is not feasible or how it came to the conclusion that the problem does not have any solution. One solution to the latter is to return to the user a sequence of simple reasoning steps that lead to inconsistency. Arc consistency is a well-known form of reasoning that can be understood by a human. We consider explanations as sequences of propagation steps of a constraint on a variable (i.e. the ubiquitous revise function in arc-consistency algorithms) that lead to inconsistency. We characterize several cases for which providing a shortest such explanation is easy: For instance when constraints are binary and variables have maximum degree two. However, these polynomial cases are tight. For instance, providing a shortest explanation is NP-hard when constraints are binary and the maximum degree is three, even if the number of variables is bounded. It remains NP-hard on trees, despite the fact that arc consistency is a decision procedure on trees. The problem is not even FPT-approximable unless the FPT $$ne $$ W[2] hypothesis is false.
{"title":"Complexity of minimum-size arc-inconsistency explanations","authors":"Christian Bessiere, Clément Carbonnel, Martin C. Cooper, Emmanuel Hebrard","doi":"10.1007/s10601-023-09360-5","DOIUrl":"https://doi.org/10.1007/s10601-023-09360-5","url":null,"abstract":"Explaining the outcome of programs has become one of the main concerns in AI research. In constraint programming, a user may want the system to explain why a given variable assignment is not feasible or how it came to the conclusion that the problem does not have any solution. One solution to the latter is to return to the user a sequence of simple reasoning steps that lead to inconsistency. Arc consistency is a well-known form of reasoning that can be understood by a human. We consider explanations as sequences of propagation steps of a constraint on a variable (i.e. the ubiquitous revise function in arc-consistency algorithms) that lead to inconsistency. We characterize several cases for which providing a shortest such explanation is easy: For instance when constraints are binary and variables have maximum degree two. However, these polynomial cases are tight. For instance, providing a shortest explanation is NP-hard when constraints are binary and the maximum degree is three, even if the number of variables is bounded. It remains NP-hard on trees, despite the fact that arc consistency is a decision procedure on trees. The problem is not even FPT-approximable unless the FPT $$ne $$ W[2] hypothesis is false.","PeriodicalId":127439,"journal":{"name":"Constraints - An International Journal","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135638325","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}
Abstract We investigate a learning decision support system for vehicle routing, where the routing engine learns implicit preferences that human planners have when manually creating route plans (or routings ). The goal is to use these learned subjective preferences on top of the distance-based objective criterion in vehicle routing systems. This is an alternative to the practice of distinctively formulating a custom vehicle routing problem (VRP) for every company with its own routing requirements. Instead, we assume the presence of past vehicle routing solutions over similar sets of customers, and learn to make similar choices. The learning approach is based on the concept of learning a Markov model, which corresponds to a probabilistic transition matrix, rather than a deterministic distance matrix. This nevertheless allows us to use existing arc routing VRP software in creating the actual routings, and to optimize over both distances and preferences at the same time. For the learning, we explore different schemes to construct the probabilistic transition matrix that can co-evolve with changing preferences over time. Our results on randomly generated instances and on a use-case with a small transportation company show that our method is able to generate results that are close to the manually created solutions, without needing to characterize all constraints and sub-objectives explicitly. Even in the case of changes in the customer sets, our approach is able to find solutions that are closer to the actual routings than when using only distances, and hence, solutions that require fewer manual changes when transformed into practical routings.
{"title":"Learn and route: learning implicit preferences for vehicle routing","authors":"Rocsildes Canoy, Víctor Bucarey, Jayanta Mandi, Tias Guns","doi":"10.1007/s10601-023-09363-2","DOIUrl":"https://doi.org/10.1007/s10601-023-09363-2","url":null,"abstract":"Abstract We investigate a learning decision support system for vehicle routing, where the routing engine learns implicit preferences that human planners have when manually creating route plans (or routings ). The goal is to use these learned subjective preferences on top of the distance-based objective criterion in vehicle routing systems. This is an alternative to the practice of distinctively formulating a custom vehicle routing problem (VRP) for every company with its own routing requirements. Instead, we assume the presence of past vehicle routing solutions over similar sets of customers, and learn to make similar choices. The learning approach is based on the concept of learning a Markov model, which corresponds to a probabilistic transition matrix, rather than a deterministic distance matrix. This nevertheless allows us to use existing arc routing VRP software in creating the actual routings, and to optimize over both distances and preferences at the same time. For the learning, we explore different schemes to construct the probabilistic transition matrix that can co-evolve with changing preferences over time. Our results on randomly generated instances and on a use-case with a small transportation company show that our method is able to generate results that are close to the manually created solutions, without needing to characterize all constraints and sub-objectives explicitly. Even in the case of changes in the customer sets, our approach is able to find solutions that are closer to the actual routings than when using only distances, and hence, solutions that require fewer manual changes when transformed into practical routings.","PeriodicalId":127439,"journal":{"name":"Constraints - An International Journal","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135688398","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 : 2023-09-01DOI: 10.1007/s10601-023-09354-3
Rémy Garcia
Floating-point numbers are used in many applications to perform computations, often without the user’s knowledge. The mathematical models of these applications use real numbers that are often not representable on a computer. Indeed, a finite binary representation is not sufficient to represent the continuous and infinite set of real numbers. The problem is that computing with floating-point numbers often introduces a rounding error compared to its equivalent over real numbers. Knowing the order of magnitude of this error is essential in order to correctly understand the behaviour of a program. Many error analysis tools calculate an over-approximation of the errors. These over-approximations are often too coarse to effectively assess the impact of the error on the behaviour of the program. Other tools calculate an under-approximation of the maximum error, i.e., the largest possible error in absolute value. These under-approximations are either incorrect or unreachable. In this thesis, we propose a constraint system capable of capturing and reasoning about the error produced by a program that performs computations with floating-point numbers. We also propose an algorithm to search for the maximum error. For this purpose, our algorithm computes both a rigorous over-approximation and a rigorous under-approximation of the maximum error. An over-approximation is obtained from the constraint system for the errors, while a reachable under-approximation is produced using a generate-and-test procedure and a local search. Our algorithm is the first to combine both an over-approximation and an under-approximation of the error. Our methods are implemented in a solver, called FErA. Performance on a set of common problems is competitive: the rigorous enclosure produced is accurate and compares well with other state-of-the-art tools.
{"title":"Floating-point numbers round-off error analysis by constraint programming","authors":"Rémy Garcia","doi":"10.1007/s10601-023-09354-3","DOIUrl":"https://doi.org/10.1007/s10601-023-09354-3","url":null,"abstract":"Floating-point numbers are used in many applications to perform computations, often without the user’s knowledge. The mathematical models of these applications use real numbers that are often not representable on a computer. Indeed, a finite binary representation is not sufficient to represent the continuous and infinite set of real numbers. The problem is that computing with floating-point numbers often introduces a rounding error compared to its equivalent over real numbers. Knowing the order of magnitude of this error is essential in order to correctly understand the behaviour of a program. Many error analysis tools calculate an over-approximation of the errors. These over-approximations are often too coarse to effectively assess the impact of the error on the behaviour of the program. Other tools calculate an under-approximation of the maximum error, i.e., the largest possible error in absolute value. These under-approximations are either incorrect or unreachable. In this thesis, we propose a constraint system capable of capturing and reasoning about the error produced by a program that performs computations with floating-point numbers. We also propose an algorithm to search for the maximum error. For this purpose, our algorithm computes both a rigorous over-approximation and a rigorous under-approximation of the maximum error. An over-approximation is obtained from the constraint system for the errors, while a reachable under-approximation is produced using a generate-and-test procedure and a local search. Our algorithm is the first to combine both an over-approximation and an under-approximation of the error. Our methods are implemented in a solver, called FErA. Performance on a set of common problems is competitive: the rigorous enclosure produced is accurate and compares well with other state-of-the-art tools.","PeriodicalId":127439,"journal":{"name":"Constraints - An International Journal","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135347779","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 : 2023-09-01DOI: 10.1007/s10601-023-09353-4
Margarita Paz Castro
{"title":"Optimization methods based on decision diagrams for constraint programming, AI planning, and mathematical programming","authors":"Margarita Paz Castro","doi":"10.1007/s10601-023-09353-4","DOIUrl":"https://doi.org/10.1007/s10601-023-09353-4","url":null,"abstract":"","PeriodicalId":127439,"journal":{"name":"Constraints - An International Journal","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135896060","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 : 2023-09-01DOI: 10.1007/s10601-023-09362-3
Jan Dreier, Sebastian Ordyniak, Stefan Szeider
Abstract The constraint satisfaction problem (CSP) is among the most studied computational problems. While NP-hard, many tractable subproblems have been identified (Bulatov 2017, Zhuk 2017) Backdoors, introduced by Williams, Gomes, and Selman (2003), gradually extend such a tractable class to all CSP instances of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a CSP instance and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021) for SAT, is a more refined distance measure, which admits the parallel utilization of different backdoor variables. Bounded backdoor size implies bounded backdoor depth, but there are instances of constant backdoor depth and arbitrarily large backdoor size. Dreier, Ordyniak, and Szeider (2022) provided fixed-parameter algorithms for finding backdoors of small depth into the classes of Horn and Krom formulas. In this paper, we consider backdoor depth for CSP. We consider backdoors w.r.t. tractable subproblems $$C_Gamma $$ CΓ of the CSP defined by a constraint language $$varvec{Gamma }$$ Γ , i.e., where all the constraints use relations from the language $$varvec{Gamma }$$ Γ . Building upon Dreier et al.’s game-theoretic approach and their notion of separator obstructions, we show that for any finite, tractable, semi-conservative constraint language $$varvec{Gamma }$$ Γ , the CSP is fixed-parameter tractable parameterized by the backdoor depth into $$C_{varvec{Gamma }}$$ CΓ plus the domain size. With backdoors of low depth, we reach classes of instances that require backdoors of arbitrary large size. Hence, our results strictly generalize several known results for CSP that are based on backdoor size.
约束满足问题(CSP)是研究最多的计算问题之一。虽然np困难，但已经确定了许多可处理的子问题(bullatov 2017, Zhuk 2017)。Williams, Gomes和Selman(2003)引入的后门，逐渐将这种可处理的类扩展到类的有界距离的所有CSP实例。后门大小提供了CSP实例和可处理类之间自然但相当粗糙的距离度量。由Mählmann、Siebertz和Vigny(2021)为SAT引入的后门深度是一种更精细的距离度量，它允许并行利用不同的后门变量。有限的后门大小意味着有限的后门深度，但也存在后门深度恒定和后门大小任意大的实例。Dreier, Ordyniak, and Szeider(2022)提供了固定参数算法，用于在Horn和Krom公式类中寻找小深度的后门。本文考虑了CSP的后门深度。我们考虑后门w.r.t.由约束语言$$varvec{Gamma }$$ Γ定义的CSP的可处理子问题$$C_Gamma $$ C Γ，即，其中所有约束都使用来自语言$$varvec{Gamma }$$ Γ的关系。基于Dreier等人的博弈论方法及其分隔障碍的概念，我们证明了对于任何有限的，可处理的，半保守的约束语言$$varvec{Gamma }$$ Γ， CSP是固定参数可处理的，由后门深度$$C_{varvec{Gamma }}$$ C Γ加上域大小参数化。使用低深度的后门，我们可以得到需要任意大尺寸后门的实例类。因此，我们的结果严格概括了基于后门大小的CSP的几个已知结果。
{"title":"CSP beyond tractable constraint languages","authors":"Jan Dreier, Sebastian Ordyniak, Stefan Szeider","doi":"10.1007/s10601-023-09362-3","DOIUrl":"https://doi.org/10.1007/s10601-023-09362-3","url":null,"abstract":"Abstract The constraint satisfaction problem (CSP) is among the most studied computational problems. While NP-hard, many tractable subproblems have been identified (Bulatov 2017, Zhuk 2017) Backdoors, introduced by Williams, Gomes, and Selman (2003), gradually extend such a tractable class to all CSP instances of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a CSP instance and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021) for SAT, is a more refined distance measure, which admits the parallel utilization of different backdoor variables. Bounded backdoor size implies bounded backdoor depth, but there are instances of constant backdoor depth and arbitrarily large backdoor size. Dreier, Ordyniak, and Szeider (2022) provided fixed-parameter algorithms for finding backdoors of small depth into the classes of Horn and Krom formulas. In this paper, we consider backdoor depth for CSP. We consider backdoors w.r.t. tractable subproblems $$C_Gamma $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>Γ</mml:mi> </mml:msub> </mml:math> of the CSP defined by a constraint language $$varvec{Gamma }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> , i.e., where all the constraints use relations from the language $$varvec{Gamma }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> . Building upon Dreier et al.’s game-theoretic approach and their notion of separator obstructions, we show that for any finite, tractable, semi-conservative constraint language $$varvec{Gamma }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> , the CSP is fixed-parameter tractable parameterized by the backdoor depth into $$C_{varvec{Gamma }}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:msub> </mml:math> plus the domain size. With backdoors of low depth, we reach classes of instances that require backdoors of arbitrary large size. Hence, our results strictly generalize several known results for CSP that are based on backdoor size.","PeriodicalId":127439,"journal":{"name":"Constraints - An International Journal","volume":"142 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135688263","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 : 2023-09-01DOI: 10.1007/s10601-023-09364-1
Felix Ulrich-Oltean, Peter Nightingale, James Alfred Walker
Abstract Many constraint satisfaction and optimisation problems can be solved effectively by encoding them as instances of the Boolean Satisfiability problem (SAT). However, even the simplest types of constraints have many encodings in the literature with widely varying performance, and the problem of selecting suitable encodings for a given problem instance is not trivial. We explore the problem of selecting encodings for pseudo-Boolean and linear constraints using a supervised machine learning approach. We show that it is possible to select encodings effectively using a standard set of features for constraint problems; however we obtain better performance with a new set of features specifically designed for the pseudo-Boolean and linear constraints. In fact, we achieve good results when selecting encodings for unseen problem classes. Our results compare favourably to AutoFolio when using the same feature set. We discuss the relative importance of instance features to the task of selecting the best encodings, and compare several variations of the machine learning method.
{"title":"Learning to select SAT encodings for pseudo-Boolean and linear integer constraints","authors":"Felix Ulrich-Oltean, Peter Nightingale, James Alfred Walker","doi":"10.1007/s10601-023-09364-1","DOIUrl":"https://doi.org/10.1007/s10601-023-09364-1","url":null,"abstract":"Abstract Many constraint satisfaction and optimisation problems can be solved effectively by encoding them as instances of the Boolean Satisfiability problem (SAT). However, even the simplest types of constraints have many encodings in the literature with widely varying performance, and the problem of selecting suitable encodings for a given problem instance is not trivial. We explore the problem of selecting encodings for pseudo-Boolean and linear constraints using a supervised machine learning approach. We show that it is possible to select encodings effectively using a standard set of features for constraint problems; however we obtain better performance with a new set of features specifically designed for the pseudo-Boolean and linear constraints. In fact, we achieve good results when selecting encodings for unseen problem classes. Our results compare favourably to AutoFolio when using the same feature set. We discuss the relative importance of instance features to the task of selecting the best encodings, and compare several variations of the machine learning method.","PeriodicalId":127439,"journal":{"name":"Constraints - An International Journal","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135737244","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 : 2023-09-01DOI: 10.1007/s10601-023-09365-0
Mohamed Sami Cherif
{"title":"Reasoning and inference for (Maximum) satisfiability: new insights","authors":"Mohamed Sami Cherif","doi":"10.1007/s10601-023-09365-0","DOIUrl":"https://doi.org/10.1007/s10601-023-09365-0","url":null,"abstract":"","PeriodicalId":127439,"journal":{"name":"Constraints - An International Journal","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135735116","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 : 2023-09-01DOI: 10.1007/s10601-023-09356-1
Pierre Talbot
{"title":"Spacetime programming: a synchronous language for constraint search","authors":"Pierre Talbot","doi":"10.1007/s10601-023-09356-1","DOIUrl":"https://doi.org/10.1007/s10601-023-09356-1","url":null,"abstract":"","PeriodicalId":127439,"journal":{"name":"Constraints - An International Journal","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894805","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 : 2023-09-01DOI: 10.1007/s10601-023-09357-0
Jordi Coll Caballero
A scheduling problem can be defined in a nutshell as the problem of determining when and how the activities of a project have to be run, according to some project requirements. Such problems are ubiquitous nowadays since they frequently appear in industry and services. In most cases the computation of solutions of scheduling problems is hard, especially when some objective, such as the duration of the project, has to be optimised. The recent performance advances on solving the problems of Boolean Satisfiability (SAT) and SAT Modulo Theories (SMT) have risen the interest in formulating hard combinatorial problems as SAT or SMT formulas, which are then solved with efficient algorithms. One of the principal advantages of such logic-based techniques is that they can certify optimality of solutions.
{"title":"Scheduling through logic-based tools","authors":"Jordi Coll Caballero","doi":"10.1007/s10601-023-09357-0","DOIUrl":"https://doi.org/10.1007/s10601-023-09357-0","url":null,"abstract":"A scheduling problem can be defined in a nutshell as the problem of determining when and how the activities of a project have to be run, according to some project requirements. Such problems are ubiquitous nowadays since they frequently appear in industry and services. In most cases the computation of solutions of scheduling problems is hard, especially when some objective, such as the duration of the project, has to be optimised. The recent performance advances on solving the problems of Boolean Satisfiability (SAT) and SAT Modulo Theories (SMT) have risen the interest in formulating hard combinatorial problems as SAT or SMT formulas, which are then solved with efficient algorithms. One of the principal advantages of such logic-based techniques is that they can certify optimality of solutions.","PeriodicalId":127439,"journal":{"name":"Constraints - An International Journal","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894804","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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