[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-156408-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-156408-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 cube-and-conquer is a central approach for tackling hard instances with SAT, but propagator-augmented solvers introduce difficulties because propagators dynamically learn and add constraints during search. The study uses SAT Modulo Symmetries (SMS) as the main test case, with an SMS symmetry-breaking propagator implemented as an IPASIR-UP propagator. Extensive experiments over 20,000 CPU hours evaluate cube-and-conquer variants on three combinatorial problems. A combined pipeline with prerun learning, cubing strategies, and algorithm configuration yields new insights, reducing total solving time by 2–10x and speeding up the hardest cubes by 2–50x.",{"@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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Study  \nMarkus Kirchweger \\#   \nAlgorithms and Complexity Group, TU Wien, Austria Tomáš Peitl \\#   \nAlgorithms and Complexity Group, TU Wien, Austria Stefan Szeider \\#   \nAlgorithms and Complexity Group, TU Wien, Austria Hai Xia 1 \\#   \nAlgorithms and Complexity Group, TU Wien, Austria  \n~~ Abstract ~~  \nParallel solving via cube-and-conquer is a key method for solving hard instances with SAT. While cube-and-conquer has proven successful for pure SAT problems, notably the Pythagorean triples conjecture, its application to SAT solvers augmented with propagators presents unique challenges as propagators learn constraints dynamically during the search.  \nWe study this problem using SAT Modulo Symmetries (SMS) as our primary test case. In our setting, the SMS symmetry-breaking propagator is an ordinary IPASIR-UP propagator; the techniques below do not rely on properties specific to symmetry breaking, except in the benchmark instantiations. Through extensive experimentation comprising over 20,000 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.  \nThe comprehensive empirical evaluation provides new insights into effective cubing strategies for propagator-based SAT solving. Our best method reduces total solving time by factors of 2-10x from improved cubing, and reduces the time for the hardest cubes by factors of 2–50x.  \n2012 ACM Subject Classification Theory of computation → Constraint and logic programming; Mathematics of computing → Graph enumeration; Mathematics of computing → Extremal graph theory  \nKeywords and phrases cube and conquer, graph search, algorithm configuration, SAT solving Digital Object Identifier 10.4230/LIPIcs.CP.2026.33  \nRelated Version Preliminary Version: [https://arxiv.org/abs/2501.17201v1](https://arxiv.org/abs/2501.17201v1)  \nSupplementary Material Software: [https://doi.org/10.5281/zenodo.14746542](https://doi.org/10.5281/zenodo.14746542)  \nFunding The authors acknowledge the support by the European Union’s Horizon 2020 research and innovation programme under the Maria Skłodowska-Curie grant agreement No. 101034440, and the support from the Austrian Science Fund (FWF), project 10.55776/P36688, 10.55776/P36420 and 10.55776/COE12 .  \n 1  Introduction  \nConflict-driven clause learning (CDCL) SAT solvers can solve instances with millions of variables and clauses [13] . However, for hard combinatorial problems, parallelization becomes essential. The cube-and-conquer technique is highly effective for such problems, famously solving the Pythagorean triples conjecture [21] .  \n1 Corresponding author  \n© Markus Kirchweger, Tomáš Peitl, Stefan Szeider, and Hai Xia;  \nlicensed under Creative Commons License CC-BY 4.0  \n32nd International Conference on Principles and Practice of Constraint Programming (CP 2026) . Editor: Nicolas Beldiceanu; Article No. 33; pp. 33:1–33:19  \nLeibniz International Proceedings in Informatics  \n Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \n33:2 Smart Cubing for Graph Search: A Comparative Study  \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, enabling efficient parallelization.  \nSymmetries are a central challenge in constraint solving, and constraint programming has developed a broad range of techniques for detecting, exploiting, and breaking them [14] . Combinatorial SAT encodings, especially of graphs, often have highly symmetric search spaces. Many isomorphic graphs satisfy the same constraints, but only one canonical representative per class needs to be checked. CDCL solvers alone cannot exploit such symmetries, and static symmetry-breaking methods","cbCaigSetJbhUETd","https://ap.wps.com/l/cbCaigSetJbhUETd","pdf",1121593,"English","# Introduction\n# Smart Cubing in Cube-and-Conquer\n## Symmetries and SMS\n## Challenges of Propagator-Based Cubing\n# Experimental Pipeline and Results","[{\"question\":\"What challenge does propagator-augmented SAT solving create for cube-and-conquer?\",\"answer\":\"Propagators dynamically introduce essential constraints during search, while standard cubing may only partition based on the original formula, missing constraints generated later.\"},{\"question\":\"Why is SAT Modulo Symmetries (SMS) used as the primary test case?\",\"answer\":\"SMS provides a practical propagator-augmented setting where symmetry-breaking constraints are learned dynamically via a custom propagator.\"},{\"question\":\"What components does the proposed pipeline include?\",\"answer\":\"It combines a prerun phase to extract learned and propagator-generated clauses, look-ahead-based cubing with multiple heuristics, and parameter tuning through algorithm configuration for the conquer solver.\"}]","Smart Cubing for Graph Search - A Comparative Study | PDF",48]