1. The Search for Systems of Diagonal Latin Squares Using the SAT@home Project Oleg Zaikin, Stepan Kochemazov Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, Russia BOINC:FAST’2015 14-18 of September, 2015, Petrozavodsk, Russia

2. 2 Latin squares

3. 3 Diagonal Latin squares

4. 4 Orthogonality

5. Pair of nonorthogonal LS These LS are partially orthogonal – in 3 cells from 9 (marked with green color). Pseudopair of LS – partially orthogonal pair of LS. Characteristics of a pseudopair – number of ordered pairs of elements, for which orthogonality condition occurs. Characteristics of this pseudopair is 3. 5 Pseudopairs

6. 6 Pseudotriples

7. design of experiments design of hardware error correcting codes cryptography 7 Applications of Latin squares

8. Determining the existence of combinatorial objects with certain properties; Enumeration of all combinatorial objects with certain properties; Combinatorial optimization - searching for combinatorial objects with a record value of certain numerical characteristics. 8 Types of combinatorial problems

9. Determining the existence of combinatorial objects with certain properties Existence of a pair of MOLS of order 10. Was solved in 1959 by a supercomputer. Existence of a pair of MODLS (here D stands for diagonal) of order 10. Was solved in 1992 (3 pairs were suggested), see paper* Existence of a triple of MOLS of order 10. Famous open problem. * Brown et al. Completion of the Spectrum of Orthogonal Diagonal Latin Squares. Lecture notes in pure and applied mathematics. 1992. Vol. 139. pp. 43–49. 9 Problems of Latin squares

10. Enumeration of all combinatorial objects with certain properties Enumeration of all LS of order 10. Was solved by McKay and Rogoyski in 1995. Number is 9982437658213039871725064756920320000. Enumeration of all DLS of order 10. Open problem. 10 Problems of Latin squares

11. Combinatorial optimization - searching for combinatorial objects with a record value of certain numerical characteristics Searching for pseudotriples of LS of order 10 with best characteristics. Current best known pseudotriple (with characteristics 91) was found by Wanless et al in 2014. Searching for pseudotriples of DLS of order 10 with best characteristics. In the paper of Brown et al in 1992 a pseudotriple with characteristics 60 was presented. 11 Problems of Latin squares

12. 60 ordered pairs of elements, for which orthogonality condition occurs for all 3 pairs of squares. Pseudotriple of DLS of order 10 by Brown et al

13. Searching for pairs of MODLS in SAT@home We made a SAT encoding for problem of searching for pairs of MODLS of order 10. For each square cell we use a set of 10 Boolean variables. Obtained CNF consist of 2000 Boolean varibles and 434440 clauses, size of file with this CNF is 10 megabytes. 13

14. Generation of tasks in SAT@home In every task 74 elements from the first square and all elements of the second square are unknown. SAT solver (in the form of client application) on a BOINC host had to find it. fixed first row variable values, unique in every task Unknown elements 0123456789

15. Generation of tasks in SAT@home We produced about 230 billions of possible variants of corresponding assignments, that do not violate any condition. We decided to process in SAT@home only first 20 million subproblems out of 230 billions (i.e. about 0.0087 % of the search space. To solve each subproblem the SAT solver Minisat 2.2 had the limit of 2600 restarts that is approximately equal to 5 minutes of work on one core of Intel Core 2 Duo E8400 processor. After reaching the limit the computations were interrupted. In one workunit downloaded by project participant there were 20 of such subproblems. This number was chosen so that one job batch can be processed in about 2 hours on one CPU core.

16. Generation of tasks in SAT@home To process 20 million subproblems (in the form of 1 million job batches) it took SAT@home about 9 months (from September 2012 to May 2013). The computations for the majority of subproblems were interrupted, but 17 subproblems were solved and resulted in 17 previously unknown pairs of orthogonal diagonal Latin squares of order 10 (we compared them with the pairs from the paper by Brown er al). The amount done by SAT@hom in this experiment of calculations is equal to 46 years of work of a PC with 4-core CPU.

17. One can easily check (even manually) that this pair of squares is a real MODLS of order 10. Hard to find and easy to check. First pair of MODLS of order 10 found by SAT@home

18. We tried to find pseudotriples for which a value of the characteristics is higher than for Brown’s pseudotriple (value of the characteristics is 60). The idea is to use pairs of MODLS of order 10 for constructing psuedotriples. For particular pair we try to find a DLS such that an obtained system of these 3 DLS (2 from the pair and 1 that was found) will be a pseudotriple with high characteristics. First approach – to generate a large number of DLS and for each of them construct 20 pseudotriples (for every known pair) and check their charecteristics. Searching for pseudotriples

19. We have tried 2 variants for generating a DLS: 1. randomly choosing values of cells elements, as a result we have found a pseudotriple with characteristics 62; 2. brute-force all combinations of values of cells elements (implemented by Alexey Zhuravlev), as a result we have found a pseudotriple with characteristics 66. Both of these algorithms were implemented with the help of MPI library and were launched on a supercomputer. Searching for pseudotriples

20. We also tried SAT approach. We constructed an CNF that encodes a problem of searching for pseudotriples of DLS of order 10. After it we made 20 CNFs by adding data about known pairs. Every such CNF encode a problem of searching for a pseudotiple based on a particular pair of MODLS. Then we launched the multithread SAT solver treengeling on each CNF – one node of supercomputer for each CNF. After several days we have found a pseudotriple with characteristics 73. Searching for pseudotriples

21. 21 New pseudotriple with characterisitcs 73 73 ordered pairs of elements, for which orthogonality condition occurs for all 3 pairs of squares.

22. One of the popular question is: “Why should we find more these objects in a volunteer computing?”. In fact, a problem of searching for new pairs of MODLS of order 10 doesn’t belong to any of 3 main classes of combinatorial problems (existence, enumeration, optimization). Howrver, if we will find more pairs of MODLS of order 10 then we will increase probability of finding new pseudotriples with high value of charecteristics. This problem belongs to the class of combinatorial optimization problems. 22 A goal for searching more pairs of MODLS of order 10

23. That is why in 2015 a new experiment aimed at searching for new pairs of MODLS of order 10 was started in SAT@home. During this experiment we have found 16 new pairs. So, recently we have 33 pairs found in SAT@home and 3 pairs found by Brown et al. 23 Goal for searching more pairs of MODLS of order 10

24. Try these 16 new pairs of MODLS of order 10 for searching new pseudotriples; Try new multithread SAT solvers from the SAT 2015 competition (sources will be available soon); Try new SAT encoding for the problem of searching for pairs of MODLS of order 10; Find new pairs of MODLS of order 10 in SAT@home; Try to prove (by SAT solvers) that for particular pair of MODLS of order 10 a psuedotriple with chareteristics more than some threshold value (for example, 90) isn’t exist. 24 Future work

25. We thank Alexander Semenov for valuable discussions/ Mikhail Posypkin, Nickolay Khrapov, Vadim Bulavintsev and Maxim Manzyuk for their help in maintainning the SAT@home project. Alexey Zhuravlev for his implementation of an brute-force algorithm of searching for DLS. Eduard Vatutin for wikipedia page about SAT@home. All the SAT@home volunteers for their participation. Thank you for your attention! 25