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The results target promise constraint satisfaction settings where strong constraints guarantee solvability and weak constraints ask for an easier solution, yielding a tractability/complexity boundary driven by algebraic properties.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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Oxford, UK Stanislav Živný \\# Ñ  \nDepartment of Computer Science, University of Oxford, UK  \n~~ Abstract ~~  \nWe give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solution over a more restricted monoid exists. As a corollary, we obtain a complexity classification for the same problem over groups.  \n2012 ACM Subject Classification Theory of computation Ñ Problems, reductions and completeness; Theory of computation Ñ Constraint and logic programming  \nKeywords and phrases constraint satisfaction, promise constraint satisfaction, equations, minions Digital Object Identifier 10.4230/LIPIcs.ICALP.2024.146  \nCategory Track B: Automata, Logic, Semantics, and Theory of Programming  \nRelated Version Full Version: [https://arxiv.org/abs/2402.08434](https://arxiv.org/abs/2402.08434) [40]  \nFunding This research was funded in whole by UKRI EP/X024431/1 . For the purpose of Open Access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission. All data is provided in full in the results section of this paper.  \n 1  Introduction  \nConstraint satisfaction problems (CSPs) form a large class of fundamental computational problems studied in artificial intelligence, database theory, logic, graph theory, and computational complexity. Since CSPs (with infinite domains) capture, up to polynomial-time Turing reductions, all computational problems [11], some restrictions need to be imposed on CSPsin order to have a chance to obtain complexity classifications. One line of work, pioneered in the database theory [36], restricts the interactions of the constraints in the instance [30, 41] .  \nAnother line of work, pioneered in [34, 26], restricts the types of relations used in the instance; these CSPs are known as nonuniform CSPs, or as having a fixed template/constraint language. Such CSPs with infinite domains capture graph acyclicity, systems of linear equations over the rationals, and many other problems [10] . Already fixed-template CSPs with finite domains form a large class of fundamental problems, including graph colourings [32], variants of the Boolean satisfiability problem, and, more generally, systems of equations over different types of finite algebraic structures. Even then, the class of finite-domain CSPs avoided a complete complexity classification for two decades despite a sustained effort.  \nIn 2017, Bulatov [20] and, independently, Zhuk [47] classified all finite-domain CSPsas either solvable in polynomial time or NP-hard, thus answering in the affirmative the Feder-Vardi dichotomy conjecture [26] . In the effort to answer the Feder-Vardi conjecture, many complexity dichotomies were established in restricted fragments of CSPs. This included conservative CSPs [19], or equations over finite algebraic structures such as semigroups, groups, and monoids [29, 35] . In particular, while systems of equations 1 over Abelian groups are solvable in polynomial time, they are NP-hard over non-Abelian groups [29] .  \n1 Some papers use the term a linear equation.  \n© Alberto Larrauri and Stanislav Živný;  \nlicensed under Creative Commons License CC-BY 4.0  \n51st International Colloquium on Automata, Languages, and Programming (ICALP 2024) .  \nEditors: Karl Bringmann, Martin Grohe, Gabriele Puppis, and Ola Svensson; Article No. 146; pp. 146:1–146:18  \nLeibniz International Proceedings in Informatics  \n Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \n146:2 Solving Promise Equations over Groups and Monoids  \nOne of the recent research directions in constraint satisfaction that has attracted a lot of attention is the area of promise CSPs (PCSPs) [3, 13 , 5] . The idea is that each constraint has two versions, a strong version and a ","cbCaii7p1J5JyLLT","https://ap.wps.com/l/cbCaii7p1J5JyLLT","pdf",877765,18,"English","# Introduction\n# Contributions","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It classifies the complexity of finding solutions to systems of equations over a fixed finite monoid, given that a solution exists over a more restricted monoid.\"},{\"question\":\"How are groups involved in the results?\",\"answer\":\"The paper derives a corresponding complexity classification for the same promise-equation solving problem over groups as a special case.\"},{\"question\":\"What is the key idea behind promise constraint satisfaction used here?\",\"answer\":\"Each constraint has a strong and a weak version: solvability is promised for the strong constraints, and the task is to satisfy the weak ones, which can be computationally easier.\"}]","Solving Promise Equations over Monoids and Groups | PDF",45]