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This work proposes improved symmetry-breaking clause construction, integrates it as preprocessing for any SAT solver, and links symmetry to unusually hard structured instances, including those derived from routing problems.",{"@graph":14,"@context":73},[15,34,56],{"@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 & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/solving-difficult-sat-instances-in-the-presence-of-symmetry-research/267456/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/solving-difficult-sat-instances-in-the-presence-of-symmetry-research/267456.png","ImageObject",300,407,{"name":42,"@type":43},"Nguyễn Văn Học","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-29","2026-09-14",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",17,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"Why do SAT instances remain difficult even when many benchmarks are easy?","Question",{"text":63,"@type":64},"SAT is NP-complete, so hard instances must exist. 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Aloul, Arathi Ramani, Igor L. Markov and Karem A. Sakallah Department ofEECS, University of Michigan, Ann Arbor 48109-2122 ffaloul,ramania,imarkov, [karem](karemg@eecs.umich.edu)[g](karemg@eecs.umich.edu)[@eecs.umich.edu](karemg@eecs.umich.edu)  \nABSTRACT  \nResearch in algorithms for Boolean satisfiability and their implementations [23, 6] has recently outpaced benchmarking efforts. Most of the classic DIMACS benchmarks [10] can be solved in seconds on commodity PCs. More recent benchmarks take longer to solve because of their large size, but are still solved in minutes [25] . Yet, small and difficult SAT instances must exist because Boolean satisfiability is NP-complete.  \nWe propose an improved construction of symmetry-breaking clauses [9] and apply it to achieve significant speed-ups over current state-ofthe-art in Boolean satisfiability. Our techniques are formulated as preprocessing and can be applied to any SAT solver without changing its source code. We also show that considerations of symmetry may lead to more efficient reductions to SAT in the routing domain.  \nOur work articulates SAT instances that are unusually difficult for their size, including satisfiable instances derived from routing problems. Using an efficient implementation to solve the graph automorphism problem [18, 20, 22], we show that in structured SAT instances difficulty maybe associated with large numbers of symmetries.  \nCategories and Subject Descriptors I.1.2 [Algorithms]: Algebraic algorithms.  \nGeneral Terms  \nAlgorithms, experimentation, verification.  \nKeywords  \nSAT, CNF, faster, search, symmetry, difficult, instances, speed-up.  \n1. INTRODUCTION  \nBoolean satisfiability (SAT) is a pivotal problem in Computer Science and has numerous applications in Design Automation that range from microprocessor verification [25] to FPGA layout [19] . A one-milliondollar prize is offered by the Clay Institute for Mathematical Sciences for a complete polynomial-time SAT solver or a proof that such an algorithm does not exist (the P-vs-NP problem) . Neither is likely to be found. Nevertheless, industrial applications motivate intensive research in SAT algorithms that quickly solve real-life instances. The fundamental framework for state-of-the-art SAT algorithms was laid out in the 1960s, but a number of recent improvements in algorithms and implementation techniques [23, 6] have led to performance breakthroughs. Most DIMACS challenge benchmarks [10] from the early 1990s are now solved in seconds on commodity PCs. Recently posted SAT benchmarks [25] take somewhat longer to solve (minutes), but that is primarily due to their enormous size (50MB+ files, etc) . With the exception of artificially constructed families of benchmarks, it appears that SAT can be solved in polynomial time “for practical purposes”.  \nIt is well known that the dominant back-track solvers, such as GRASP [23] and CHAFF [6] do not perform well on randomly-created 3-SAT in  \nPermission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. DAC 2002 June 10-14, 2002, New Orleans, Louisiana, USA  \nCopyright 2002 ACM 1-58113-297-2/01/0006(1-58113-461-4) ... $5.00.  \nstances with 􀀙 4 : 3 clauses per variable. However, such instances do not arise in Design Automation because application-derived SAT instances are typically structured. Attempts to explain the ease of structured instances were successful for certain applications [21], and generic ways to exploit certain types of structure were proposed [1] .  \nOur work addresses both benchmarking and algorithmic aspects of SAT research. Given the excellent per","cbCaif0t83lxiyTH","https://ap.wps.com/l/cbCaif0t83lxiyTH","pdf",283784,"English","# Introduction\n## SAT benchmarking and instance difficulty\n## Symmetry and symmetry-breaking in SAT\n## Routing-related SAT and graph automorphism","[{\"question\":\"Why do SAT instances remain difficult even when many benchmarks are easy?\",\"answer\":\"SAT is NP-complete, so hard instances must exist. While classic DIMACS benchmarks are easy, newer ones are hard largely because of size, and difficult small instances are still expected to exist.\"},{\"question\":\"What is the main contribution of the proposed approach?\",\"answer\":\"The work introduces an improved construction of symmetry-breaking clauses and applies it to obtain significant speed-ups over state-of-the-art SAT solvers.\"},{\"question\":\"How can the technique be used with existing SAT solvers?\",\"answer\":\"The techniques are formulated as preprocessing, so they can be applied to any SAT solver without modifying its source code.\"}]","Solving Difficult SAT Instances in the Presence of Symmetry - Research | PDF",15]