Automatic Generation of Constraints for Partial Symmetry Breaking
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1 Automatic Generation of Constraints for Partial Symmetry Breaking Karen Petrie and Christopher Jefferson
2 Overview How to break symmetries. How to find symmetries. How to choose which symmetries to break.
3 Magic Square = 15 = 15 = 15 = 15 = 15= 15 = 15 = 15
4 Magic Square = = 15 = = 15 = 15= 15 = 15 = 15
5 Magic Square
6 Magic Square
7 Magic Square
8 Magic Square
9 Magic Square A B C D E F G H I G D A H E B I F C ABCDEFGHI lex GDAHEBIFC
10 Magic Square
11 Magic Square X
12 Magic Square There are 8 symmetries in total Identity 90 left right X-axis swap Y-axis swap Major diagonal swap Minor diagonal swap
13 Crawford Ordering Add one lexicographic ordering constraint for every member of the symmetry group. Breaks all symmetry.
14 Crawford Ordering Problems often have a huge number of symmetries A 10x10 matrix with ``row and column'' symmetry has 13,168,189,440,000 symmetries.
15 Partial Symmetry Breaking Using a subset of the symmetry breaking constraints breaks some symmetry. Need to decide: How many constraints to use. Exactly which to use.
16 Now For Something Completely Different... We need to know the symmetries of a problem to apply any of this. Wouldn't it be great if we could detect them automatically?
17 Magic Square
18 Magic Square
19 Magic Square
20 Magic Square
21 Magic Square
22 Symmetry Generation Nauty and Saucy will find the symmetries of a graph. They both use a similar algorithm to find a set of generators for the group.
23 Magic Square Identity X-axis swap X 90 left right R Y-axis swap Diag swap 1 Diag swap 2
24 Magic Square Identity XX X-axis swap X 90 left R Y-axis swap XRR 180 RR Diag swap 1 XR 90 right RRRR Diag swap 2 XRRR
25 Choosing Symmetry Breaking Constraints Now we have the group of symmetries. Represented as a small set of generators. How do we choose a subset to generate symmetry breaking constraints from?
26 Nauty Constraints Using the symmetries Nauty generates is the simplest option.
27 Random Symmetries 1e+10 1e+09 Random Nauty Basic Stabiliser Reduced ArityOne 1e+08 1e+07 x 1e Number of Constraints
28 Investigating the Nauty Constraints They appear to perform quite well, at least compared to a random set of symmetries. Let's figure out why!
29 Generators The symmetries that Nauty creates are generators for the group. As they ``represent'' all the group, should they create a good set of constraints.
30 Generators Sets of generators are no better, or worse, than an arbitrary set of symmetries. Checked over a range of problems.
31 Generators Some common sets of symmetry breaking constraints are generators. Most generator sets are bad There are equally good non-generator sets.
32 Generators
33 Generators
34 Generators
35 Why is Nauty so Successful? At this point we had to go back to the drawing board. Why is Nauty so successful?
36 Complete Symmetry Group We already know how to break this symmetry completely, with a linear number of constraints! But analysing this group can give us guidance.
37 Permutations A B C D E F G H
38 Permutations A B C D E F G H ABCDEFGH lex B...
39 Permutations A B C D E F G H ABCDEFGH lex AC...
40 Permutations A B C D E F G H ABCDEFGH lex ABD...
41 Permutations A B C D E F G H ABCDEFGH lex B... ABCDEFGH lex AC... ABCDEFGH lex ABD... ABCDEFGH lex ABCE... ABCDEFGH lex ABCDF... ABCDEFGH lex ABCDEG.. ABCDEFGH lex ABCDEFHG
42 Stabiliser Chains Internally, Nauty generates a `Stabiliser Chain' How Nauty represents the set of generators as it produces them.
43 Permutations A B C D E F G H ABCDEFGH lex B... ABCDEFGH lex AC... ABCDEFGH lex ABD... ABCDEFGH lex ABCE... ABCDEFGH lex ABCDF... ABCDEFGH lex ABCDEG.. ABCDEFGH lex ABCDEFHG
44 Random Permutations Let us consider a random permutation of n variables. There is a 1/n chance it maps the first variable to itself. There is a 1/n(n-1) chance it maps the first and second variable to themselves. We need the last symmetry exactly.
45 Magic Square A B C D E F G H I G D A H E B I F C ABCDEFGHI lex GDAHEBIFC
46 Magic Square A B C D E F G H I G D A H E B I F C ABCDEFGHI lex GDAHEBIFC
47 Magic Square A B C D E F G H I A D G B E H C F I ABCDEFGHI lex ADGBEHCFI
48 Magic Square A B C D E F G H I A D G B E H C F I ABCDEFGHI lex ADGBEHCFI
49 First Variable Pair ABCDEFGHI lex GDAHEBIFC A < G: Constraint true! A > G: Constraint false! A = G: Have to look at other variables.
50 First Variable Pair Only 1/d assignments are not decided by the first variable pair. In practice the symmetry breaking constraints and problem constraints are not so independent.
51 First Variable Pairs We want lots of different first variable pairs. Ideally we want a set of constraints which logically implies all possible first variable pairs. Nauty almost does this!
52 Stabiliser Chains Buried in Nauty is an optimisation condition which looks like: A < B We can weaken this to: A B And Nauty produces constraints which imply all the First Variable Pairs. And strictly more symmetries.
53 1e+10 1e+09 Random Nauty Basic Stabiliser Reduced ArityOne 1e+08 1e+07 1e Number of Constraints
54 Future Study more problems and more groups. Variables after the FVP still matter, so how should we choose them? Better understanding of which constraints to choose to get maximal coverage.
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