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Generic SBDD using Computational Group Theory

Generic SBDD using Computational Group Theory . Ian Gent St Andrews Warwick Harvey Imperial College Tom Kelsey St Andrews Steve Linton St Andrews. Conclusions . Two key novelties Entirely generic implementation of SBDD

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Generic SBDD using Computational Group Theory

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  1. Generic SBDD using Computational Group Theory Ian Gent St Andrews Warwick Harvey Imperial College Tom Kelsey St Andrews Steve Linton St Andrews

  2. Conclusions • Two key novelties • Entirely generic implementation of SBDD • user defines the group, does not write a dominance checker • Algebraic algorithm for dominance checking • searches given definition of an arbitrary group • Implementation provided in GAP-ECLiPSe • Groups up to size 1036 • compared to 109 for SBDS in GAP/ECLiPSe • Future work • Exploit special features of commonly occurring groups Generic SBDD

  3. Outline of Talk • Conclusions • Symmetry Breaking, SBDD, Dominance Checking • Groups and GAP • Generic Dominance Check • Discussion • Conclusions Generic SBDD

  4. Symmetry Breaking • Many CSPs have symmetry • n-queens, colouring, golfers’ problem, … • Breaking symmetry reduces search • avoids exploring equivalent states • Three main approaches to symmetry breaking • reformulate the problem • add constraints before search • adapt search algorithm to break symmetry • SBDD takes the last of these approaches Generic SBDD

  5. SBDD • Symmetry Breaking by Dominance Detection • Fahle, Schamberger, Sellmann, 2001 • Foccaci, Milano, 2001 • prefigured by Brown, Finkelstein, Purdom, 1988 • Do not search a node if you have searched its equivalent before • check before entering a node Generic SBDD

  6. Dominance Check 1. Deduce consequences of X≠a, Y=b, Z≠c, W ≠d 2. Is X=a equivalent to anything deduced? 3. Is Y=b, Z=c equivalent to anything deduced? 4. Is Y=b, W=d equivalent to anything deduced? Generic SBDD

  7. Generic Dominance Check • Two existing approaches to dominance checking • hand code checking procedure • express check as a constraint problem • Dominance check works only for a family of instances • e.g. golfers’ problems, BIBD instances • We have written a generic dominance checker • works for any constraint problem • user has only to define the group acting on the CSP Generic SBDD

  8. Generators and Groups • Consider simple problem • A3 + B3 + C3 + D3 = E3 + F3 + G3 • A…G :: 1..50 • Given any solution, we can permute the values of ABCD and of EFG, independently. • (A B C D) is permutation cycling A through D • (A D) swaps A and D • (E F G) cycles E through G • (E G) swaps E and G • These four permutations generate the group of different ways to construct one solution from another. • there are 4!3! = 144 group elements Generic SBDD

  9. GAP and GAP-ECLiPSe • GAP is world leading computational algebra package • accepts generators and computes using the whole group • e.g. we pass four elements (A B C D), (A D), (E F G), (E G) • GAP computes with full group of size 144 • 144 is unusually small • number of generators is at worst logarithmic in size of group • GAP-ECLiPSe • define generating elements and pass to GAP • constraint problem searched in ECLiPSe • dominance check written in GAP • could be done in any pair of Algebra/Constraint packages Generic SBDD

  10. Generic Dominance Check • failsets • sets of decisions at failed nodes • {X=a}, {Y=b,Z=c}, {Y=b,W=d} • pointset • set of assigned variables at current node • {Y=b,X=b,V=c} • includes propagated values • assume X=b, V=c set by inference • It is a heuristic choice to have failsets as small as possible, pointset as big as possible. Generic SBDD

  11. Dominance Check as Search • Dominance check is an algebraic search • Aside: anyone who has written a dominance check has written a computational algebra program! • Seek group element g and failset S s.t. • S g Pointset • Search process is essentially straightforward • but we omit most algebraic details here • critical to performance by several orders of magnitude Generic SBDD

  12. Dominance Check as Search • Recursive backtracking search • work through failset • find ways of mapping first element of failset to elements of pointset: try these in turn • recurse for second element, fixing the mapping of the first element • GAP allows us to make computation efficient • work in stabilizers, subgroups fixing work done so far • use transversals, the places that elements can map to • use Schreier vectors for efficiency Generic SBDD

  13. Dominance Check as Search • is there g, S with S g Pointset ? • S in failsets: {X=a}, {Y=b,Z=c}, {Y=b,W=d} • Pointset: {Y=b,X=b,V=c} • Use the fact that Y=b shared between second two failsets • Can we map X=a into Pointset? • no! • Can we map Y=b into Pointset? • Yes! • Y=b  Y=b • Use Stabilizer of Y=b • Can we map Z=c into {X=b,V=c}? • no! • Can we map W=d into {X=b, V=c}? • Yes! • {Y=b,W=d}  {Y=b, V=c} • Dominance detected so backtrack Generic SBDD

  14. Inference from Dominance • Dominance can deduce domain removals • Where a domain value would make dominance succeed, it can be removed • Already known (Fahle et al, Brown et al) • Again generically implemented • Work heuristically • i.e. only report removals we find easily • Petrie [SymCon03] shows this can increase search compared to SBDS, which finds all removals Generic SBDD

  15. Overview of Results • Can solve problems with very large groups in very generic manner • Main data on BIBD’s • solve up to <7,21,6,2,1>, group 1021 in few secs • solve up to <13,26,6,3,1>, group 1036 in few hrs • count number of symmetrically distinct solutions • compare with generic SBDS in GAP-ECLiPSe • only practical for <6,10,5,3,2>, group 109 • Competitive CSP method for solving BIBD’s • special purpose BIBD methods much faster • Hits a limit at moderate sized groups such as 1036 Generic SBDD

  16. Problems? Solutions? • GAP search can be prohibitive • e.g. in <13,26,6,3,1> BIBD, group size 1036 • 2053 dominance checks take 59,000 cpu secs • Should be able to exploit structure of group in general ways to reduce this in many cases • Constraint programmers don’t know groups • We have supplied macros to express groups easily • Harvey, Kelsey, Petrie (SymCon03) • McDonald (SymCon03) Generic SBDD

  17. Conclusions • Two key novelties • Entirely generic implementation of SBDD • user defines the group, does not write a dominance checker • Algebraic algorithm for dominance checking • searches given definition of an arbitrary group • Implementation provided in GAP-ECLiPSe • Groups up to size 1036 • compared to 109 for SBDS in GAP/Eclipse • Future work • Exploit special features of commonly occurring groups Generic SBDD

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