The Quest for Perfect and Compact Symmetry Breaking for Graph Problems

The Quest for Perfect nd Compct Symmetry Breking for Grph Prolems Mrijn J.H. Heule SYNASC Septemer 25, 2016 1/19

Stisfiility (SAT) solving hs mny pplictions... forml verifiction grph theory ioinformtics trin sfety plnning comintorics cryptogrphy rewrite termintion encode SAT solver decode..., ut SAT solving my struggle in the presence of symmetries 2/19

Breking Grph Symmetries Computing Compct & Perfect Symmetry Breking Logic Minimiztion Stisfiility Solving Rndom Proing Results Conclusions nd Future Work 3/19

Breking Grph Symmetries 4/19

Brek Symmetries for Grph Existence Prolems A grph existence prolem sks whether there exists undirected grph with certin property. For exmple, does every grph of six vertices hve clique or co-clique of size 3? (Known s Rmsey numer 3) Grph existence prolems re hrd for SAT solvers due to the symmetries. Consider ll grphs with three vertices: c c c c c c c c We cn perfectly rek ll symmetries y eliminting ll ut one grph from ech isomorphism clss. For exmple, eliminting grphs 3 to 6: ( c c) ( c c) ( c c) ( c c) This cn e simplified to ( c) (c c). 5/19

Existing Techniques Brek Symmetries Prtilly Existing symmetry-reking methods constrin the djcency mtrix: qud Row i less thn or equl to row i + 1, while ignoring columns i, i + 1 cuic Row i less thn or equl to row j (i < j), while ignoring columns i, j redundncy rtio 50 40 30 20 10 0 qud cuic 1 2 3 4 5 6 7 8 9 10 11 12 size of the grph Redundncy rtio: verge numer of grphs per isomorphism clss 6/19

Computing Compct & Perfect Symmetry Breking 7/19

Logic Minimiztion Method Perfect isoltor: perfect symmetry-reking predicte Compute perfect isoltor s follows: 1. Choose cnonicl set of grphs, i.e., exctly one grph out of ech isomorphism clss; 2. Convert the cnonicl set into cluses (Tseitin encoding); 3. Reduce the size of the cluses vi logic minimiztion. For exmple for grphs with three vertices: 1. Cnon: c c c c 2. Tseitin encoding results in formul with 13 cluses (independent on cnon). 3. Cn e reduced to two cluses (dependent on cnon). 8/19

Logic Minimiztion Sizes nd Runtimes Severl tools exist to generte cnonicl set: Nuty y y Brendn McKy (1981) Bliss y Tommi Junttil nd Petteri Kski (2007) Severl tools exist to minimize given logicl formul: Espresso y y Roert Bryton (1984) Bic y Alexey Igntiev (2015) Best results with Nuty nd Bic (size in cues / cluses): k 2 3 4 5 6 7 8 P DNF 2 4 11 34 156 1, 044 12, 346 P CNF 3 13 67 341 2, 341 21, 925 345, 689 P simp 0 2 9 24 77 311 > 1, 839 9/19

SAT Solving Method The prior method required cnonicl set s input. However, the numer of choices for the cnonicl set is exponentil. Only some choices my e reducile to compct predicte. As n lterntive pproch, we trnslte the prolem into SAT: Formul F k,m expresses the SAT encoding of the existence of perfect isoltor for k vertices using m cluses. All m cluses re stisfied y grphs in the cnonicl set; Ech non-cnonicl grph flsifies t lest one cluse; These formuls re huge: O(2 E m E ) with E = k2 k 2. Using the SAT pproch, the optiml perfect symmetry reking for grphs of size k cn e computed: Find m such tht F k,m 1 is unstisfile, while F k,m is stisfile. 10/19

SAT Solving Sizes nd Runtimes Formul F k,m expresses the SAT encoding of the existence of n isoltor for k vertices using m cluses. Two top-tier solvers: glucose (G) nd treengeling (T) formul result vriles cluses est runtime F 4,6 UNSAT 756 2, 458 0.18 (G) F 4,7 SAT 861 2, 827 0.01 (G) F 5,11 UNSAT 14, 480 54, 756 3, 510.36 (T) F 5,12 SAT 15, 609 59, 281 102.69 (G) Runtimes re in wll clock seconds on qud core Intel Xeon E31280 CPU. All formuls with k 6 ppered too hrd: i.e, unsolvle in 24 hours using prllel solver running on 24 cores. 11/19

Rndom Proing Method Logic minimiztion results in lrge perfect isoltors, s existing cnonicliztion lgorithms produce poor cnonicl sets. The SAT method results in optiml isoltors, ut doesn t scle. Our third pproch is sed on rndom proing: 1. All grphs of size k re ctive nd the isoltor is empty. 2. Rnk ll potentil cluses tht cn e dded to the isoltor. The more ctive grphs tht re flsified y cluse, the higher its rnk. Ties re roken rndomly. 3. Rndomly dd single cluse to the isoltor with proility P(r) = 2 r, with r eing the cluse rnk. 4. Terminte if the isoltor is perfect. Otherwise to go 2. 12/19

Rndom Proing Sizes nd Runtimes Rndom proing cn e improved y running multiple rounds: In round i + 1 we pick the smllest isoltors of round i nd forced the first 10i cluses from those isoltors. Below two proility plots: (left) the results of 2 rounds on n = 6 with 400,000 proes per round, nd (right) the results of 4 rounds on n = 7 with 80,000 proes per round. proility 10 1 10 2 10 3 10 4 10 5 round 1 round 2 10 1 10 2 10 3 10 4 round 1 round 2 round 3 round 4 10 6 30 35 40 45 50 55 60 65 numer of cluses 10 5 120 140 160 180 200 220 numer of cluses 13/19

Results 14/19

Optiml Isoltors in CNF Vrile xy denotes whether n edge from node x to y exists. For exmple c = 0 mens there is no edge from node to c. P 3 := ( c) (c c) P 4 := (d d) (d c) (cd c) ( c) (c c) ( d cd) (c d d) P 5 := (d d) (d c) (cd c) (c d) (e ce) (e e) ( d cd) (e de e) (d ce de) ( cd de) (c d ce) (ce e c) 15/19

Visuliztion of Optiml Isoltors for n {3, 4} Two cnonicl grphs re connected with n rc if they differ in exctly one edge. The rrow points from the cnonicl grph without the edge to the one with the edge. P 3 : c P 4 : d c 16/19

Visuliztion of Optiml Isoltor for n = 5 P 5 : e d c 17/19

Conclusions nd Future Work 18/19

Conclusions nd Future Work Conclusions: We presented three methods to compute perfect symmetry-reking predicted for grph prolems. Optiml isoltors re compct, t lest for smll grphs. Existing cnonicl lel lgorithms do not llow the construction of smll isoltors. Future work: How to compute optiml isoltors for medium grphs? How to construct compct isoltors for lrge grphs? Cn grphs symmetries e perfectly roken using polynomil-sized predictes? 19/19