[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-156405-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-156405-en":131},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","smart-cubing-for-graph-search-a-comparative-study","Smart Cubing for Graph Search - A Comparative Study","","Parallel solving via cube-and-conquer is a key method for scaling SAT solvers to hard instances. While cube-and-conquer has succeeded on pure SAT tasks, extending it to SAT solving with propagators introduces challenges because propagators learn constraints dynamically during the search. This work studies the issue using SAT Modulo Symmetries (SMS), where a symmetry-breaking propagator learns constraints that eliminate isomorphic graphs. Extensive experiments over 10,000 CPU hours compare cube-and-conquer variants and show 2–3x speedups and an additional 1.5–2x gain on harder cases.",{"@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 & 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Comparative Study * Markus Kirchweger, Hai Xia, Tomˇs Peitl, and Stefan Szeider  \nAlgorithms and Complexity Group  \nTU Wien, Vienna, Austria  \n[mk,hxia,peitl,sz]@[ac.tuwien.ac.at](ac.tuwien.ac.at)  \nAbstract  \nParallel solving via cube-and-conquer is a key method for scaling SAT solvers to hard instances. While cube-and-conquer has proven successful for pure SAT problems, notably the Pythagorean triples conjecture, its application to SAT solvers extended with propagators presents unique challenges, as these propagators learn constraints dynamically during the search.  \nWe study this problem using SAT Modulo Symmetries (SMS) as our primary test case, where a symmetry-breaking propagator reduces the search space by learning constraints that eliminate isomorphic graphs. Through extensive experimentation comprising over 10,000 CPU hours, we systematically evaluate different cube-and-conquer variants on three well-studied combinatorial problems. Our methodology combines prerun phases to collect learned constraints, various cubing strategies, and parameter tuning via algorithm configuration and LLM-generated design suggestions.  \nThe comprehensive empirical evaluation provides new insights into effective cubing strategies for propagator-based SAT solving, with our best method achieving speedups of 2-3x from improved cubing and parameter tuning, providing an additional 1.5-2x improvement on harder instances.  \n1 Introduction  \nPropositional satisfiability (SAT) solvers based on conflict-driven clause learning can solve huge instances with millions of variables and clauses [Fichte et al., 2023a] . However, for hard instances, particularly in combinatorial problems, parallelization becomes necessary. The cube-and-conquer technique has proven highly effective for such problems, most notably in resolving the Pythagorean triples conjecture [Heule et al., 2016] .  \nIn cube-and-conquer, a look-ahead solver first partitions the search space into disjoint subproblems via cubes (partial assignments), which are then solved independently by CDCL solvers. This independence enables efficient parallel solving.  \nWhen encoding combinatorial problems into SAT, particularly those involving graphs, we often encounter highly symmetric search spaces. Many mutually isomorphic graphs satisfy the same constraints, but a solver needs to check only one representative, the canonical element, from each isomorphism class. Standard CDCL solvers cannot leverage these symmetries, and static symmetry breaking methods cannot break all symmetries [Codish et al., 2019] .  \nSAT Modulo Symmetries (SMS) [Kirchweger and Szeider, 2021; Kirchweger and Szeider, 2024] addresses this limitation through dynamic symmetry breaking, using a custom propagator that learns symmetry-breaking predicates during the search. SMS has been successfully applied to several  \n*Research supported by the Austrian Science Fund (FWF) within the projects 10.55776/36688 and 10.55776/COE12 .  \nhard combinatorial problems, including problems from extremal combinatorics on on graphs, hypergraphs, matroids, and clause sets 1.  \nWhile one could apply standard cube-and-conquer to any SAT-based system with propagators, this approach yields poor performance. The key challenge is that propagators learn constraints dynamically which are not present in the initial formula. In SMS, for instance, the cubing solver working only with the input formula effectively partitions the space of all graphs rather than the space of canonical graphs. Other propagator-based systems face similar issues when the propagators contribute essential constraints to reduce the search space.  \nOur contribution is a systematic study of cube-and-conquer methods adapted for SAT solving with propagators, using SMS as the primary test case. Following a carefully designed experimental methodology, we evaluate several cubing strategies and investigate effective pa","cbCaikZAmgfHXBI2","https://ap.wps.com/l/cbCaikZAmgfHXBI2","pdf",443526,15,"English","# Introduction\n## Cube-and-conquer and symmetry in SAT\n## Problem setting with dynamic propagators (SMS)\n# Preliminaries\n## Notation and background","[{\"question\":\"Why does cube-and-conquer become challenging for SAT solving with propagators?\",\"answer\":\"Because propagators learn essential constraints dynamically during the search, which are not present in the initial formula, causing poor space partitioning if cubing is done only on the input.\"},{\"question\":\"What role does SAT Modulo Symmetries (SMS) play in the study?\",\"answer\":\"SMS serves as the primary test case, using a dynamic symmetry-breaking propagator that learns predicates to eliminate isomorphic graphs during search.\"},{\"question\":\"What performance improvements do the authors report?\",\"answer\":\"The prerun and cubing pipeline yields 2–3x speedups over standard SMS, and parameter tuning adds another 1.5–2x improvement, especially on harder instances.\"}]","Smart Cubing for Graph Search - A Comparative Study | PDF",38]