Problem Solving with Max SAT Original Problem Optimal

Problem Solving with Max. SAT Original Problem Optimal Solution Encode Max. SAT Problem Max. SAT Solver Decode Max. SAT Model • Planning / Scheduling • Max. Cut, Max. Clique, Combinatorial Auction • Package upgradability 2

Overview l Weighted Partial Max. SATソルバーの分類 l Qw. Max. SAT: Q-dai weighted Max. SAT solver Ø 解法の概要，使い方，評価 l おわりに 3

Max. SATソルバーの分類 l Branch and Bound Ø akmaxsat, Clone, Inc. WMax. Satz, IUT_BCMB, Mini. Max. SAT l SAT-based • satisfiability-based Ø QMax. SAT, Sat 4 j-Max. SAT, clasp-Max. Sat • unsatisfiability-based Ø PM 2, wbo, pwbo, MSUn. Core, WPM 1 l 他のソルバーの問題形式に変換 new Ø ILP (→ CPLEX)，SCIP-maxsat (→ SCIP) l Portfolio, ハイブリッド new Ø ISAC+，Max. HS 5

QMax. Sat: Q-dai Max. SAT Solver v. Partial Max. SAT (PM) Category v Version 0. 1 : 1 st in Industrial subcategory (2010) 2 nd in Crafted subcategory (2010) v Version 0. 11 : 3 rd in Industrial subcategory (2011) 2 nd in Crafted subcategory (2011) v Version 0. 4: 1 st in Industrial subcategory (2011) 5 th in Crafted subcategory (2011) v Version 0. 21: 4 th in Industrial subcategory (2012) 1 st in Crafted subcategory (2012) v Version 0. 21 -g 2: 1 st in Industrial subcategory (2012) 4 th in Crafted subcategory (2012) v Version 0. 21 -g 2 -mt: 2 nd in Industrial subcategory (2013) 10 th in Crafted subcategory (2013) 6

Qw. Max. Sat : 性能 card warn comp Crafted Industrial 全 340問 全 396問 0 211 75. 31 10 244 114. 85 11 bail 参考 Max. SAT 2013 計測中 0 Max. HS 330 5. 92 243 53. 67 ISAC+-wpms 323 12. 85 348 51. 69 解けた問題数（制限時間３０分）／平均時間（秒） 9

実行環境 l Qw. Max. SAT l CPU: Core i 7 -2600 (3. 40 GHz) l Memory: 8 GB l OS: Ubuntu 12. 04 LTS l Max-SAT 2013 l CPU: Xeon E 7 -8837@2. 67 GHz l Memory: 3. 5 GB l OS: SUSE Linux Enterprise Server 11 SP 1 制限時間：３０分 10

参考文献 〔QMax. SAT〕 Koshimura, M. , Zhang, T. , Fujita, H. , & Hasegawa, R. (2012). QMax. SAT: A Partial Max-SAT Solver system description. Journal on Satisfiability, Boolean Modeling and Computation, 8, 95 -100. 〔提携構造形成問題のMax. SAT符号化〕 Liao, X. , Koshimura, M. , Fujita, H. , & Hasegawa, R. (2012, November). Solving the Coalition Structure Generation Problem with Max. SAT. In Tools with Artificial Intelligence (ICTAI), 2012 IEEE 24 th International Conference on (Vol. 1, pp. 910 -915). IEEE. 〔Max. SAT符号化のTIPS〕 越村 三幸，廖 暁鵑，藤田 博，長谷川 隆三． Max. SATの一拡張について， 第 11回情報科学技術フォーラム(FIT 2012)， F-028，2012年 〔Joost P. Warners 1998〕 Warners, J. P. (1998). A linear-time transformation of linear inequalities into conjunctive normal form. Information Processing Letters, 68(2), 63 -69. 〔Olivier Bailleux 2003〕 Bailleux, O. , & Boufkhad, Y. (2003, January). Efficient CNF encoding of Boolean cardinality constraints. In Principles and Practice of Constraint Programming–CP 2003 (pp. 108 -122). Springer Berlin Heidelberg. 〔Roberto Asín 2009〕 Asín, R. , Nieuwenhuis, R. , Oliveras, A. , & Rodríguez-Carbonell, E. (2009). Cardinality networks and their applications. In Theory and Applications of Satisfiability Testing-SAT 2009 (pp. 167 -180). Springer Berlin Heidelberg. 14