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Phase transition behaviour Toby Walsh Dept of CS University of York Outline What have phase transitions to do with computation? How can you observe such behaviour in your favourite problem? Is it confined to random and/or NP-complete problems? Can we build better algorithms using knowledge about phase transition behaviour? What open questions remain? 2 Health warning To aid the clarity of my exposition, credit may not always be given where it is due Many active researchers in this area: Achlioptas, Chayes, Dunne, Gent, Gomes, Hogg, Hoos, Kautz, Mitchell, Prosser, Selman, Smith, Stergiou, Stutzle, … Walsh 3 Before we begin A little history ... Where did this all start? At least as far back as 60s with Erdos & Renyi thresholds in random graphs Late 80s pioneering work by Karp, Purdom, Kirkpatrick, Huberman, Hogg … Flood gates burst Cheeseman, Kanefsky & Taylor’s IJCAI-91 paper In 91, I has just finished my PhD and was looking for some new research topics! 5 Phase transitions Enough of the history, what has this got to do with computation? Ice melts. Steam condenses. Now that’s a proper phase transition ... An example phase transition Propositional satisfiability (SAT) does a truth assignment exist that satisfies a propositional formula? NP-complete (x1 v x2) & (-x2 v x3 v -x4) x1/ True, x2/ False, ... 3-SAT formulae in clausal form with 3 literals per clause remains NP-complete 7 Random 3-SAT Random 3-SAT sample uniformly from space of all possible 3-clauses n variables, l clauses Which are the hard instances? around l/n = 4.3 What happens with larger problems? Why are some dots red and others blue? 8 Random 3-SAT Varying problem size, n Complexity peak appears to be largely invariant of algorithm backtracking algorithms like Davis-Putnam local search procedures like GSAT What’s so special about 4.3? 9 Random 3-SAT Complexity peak coincides with solubility transition l/n < 4.3 problems underconstrained and SAT l/n > 4.3 problems overconstrained and UNSAT l/n=4.3, problems on “knifeedge” between SAT and UNSAT 10 “But it doesn’t occur in X?” X = some NP-complete problem X = real problems X = some other complexity class Little evidence yet to support any of these claims! 11 “But it doesn’t occur in X?” X = some NP-complete problem Phase transition behaviour seen in: TSP problem (decision not optimization) Hamiltonian circuits (but NOT a complexity peak) number partitioning graph colouring independent set ... 12 “But it doesn’t occur in X?” X = real problems No, you just need a suitable ensemble of problems to sample from? Phase transition behaviour seen in: job shop scheduling problems TSP instances from TSPLib exam timetables @ Edinburgh Boolean circuit synthesis Latin squares (alias sports scheduling) ... 13 “But it doesn’t occur in X?” X = some other complexity class Ignoring trivial cases (like O(1) algorithms) Phase transition behaviour seen in: polynomial problems like arc-consistency PSPACE problems like QSAT and modal K ... 14 “But it doesn’t occur in X?” X = theorem proving Consider k-colouring planar graphs k=3, simple counter-example k=4, large proof k=5, simple proof (in fact, false proof of k=4 case) 15 Locating phase transitions How do you identify phase transition behaviour in your favourite problem? What’s your favourite problem? Choose a problem e.g. number partitioning dividing a bag of numbers into two so their sums are as balanced as possible Construct an ensemble of problem instances n numbers, each uniformly chosen from (0,l ] other distributions work (Poisson, …) 17 Number partitioning Identify a measure of constrainedness more numbers => less constrained larger numbers => more constrained could try some measures out at random (l/n, log(l)/n, log(l)/sqrt(n), …) Better still, use kappa! (approximate) theory about constrainedness based upon some simplifying assumptions e.g. ignores structural features that cluster solutions together 18 Theory of constrainedness Consider state space searched see 10-d hypercube opposite of 2^10 truth assignments for 10 variable SAT problem Compute expected number of solutions, <Sol> independence assumptions often useful and harmless! 19 Theory of constrainedness Constrainedness given by: kappa= 1 - log2(<Sol>)/n where n is dimension of state space kappa lies in range [0,infty) kappa=0, <Sol>=2^n, kappa=infty, <Sol>=0, kappa=1, <Sol>=1, under-constrained over-constrained critically constrained phase boundary 20 Phase boundary Markov inequality prob(Sol) < <Sol> Now, kappa > 1 implies <Sol> < 1 Hence, kappa > 1 implies prob(Sol) < 1 Phase boundary typically at values of kappa slightly smaller than kappa=1 skew in distribution of solutions (e.g. 3-SAT) non-independence 21 Examples of kappa 3-SAT kappa = l/5.2n phase boundary at kappa=0.82 3-COL kappa = e/2.7n phase boundary at kappa=0.84 number partitioning kappa = log2(l)/n phase boundary at kappa=0.96 22 Number partition phase transition Prob(perfect partition) against kappa 23 Finite-size scaling Simple “trick” from statistical physics around critical point, problems indistinguishable except for change of scale given by simple power-law Define rescaled parameter gamma = kappa-kappac . n^1/v kappac estimate kappac and v empirically e.g. for number partitioning, kappac=0.96, v=1 24 Rescaled phase transition Prob(perfect partition) against gamma 25 Rescaled search cost Optimization cost against gamma 26 Easy-Hard-Easy? Search cost only easy-hard here? Optimization not decision search cost! Easy if (large number of) perfect partitions Otherwise little pruning (search scales as 2^0.85n) Phase transition behaviour less well understood for optimization than for decision sometimes optimization = sequence of decision problems (e.g branch & bound) BUT lots of subtle issues lurking? 27 Algorithms at the phase boundary What do we understand about problem hardness at the phase boundary? How can this help build better algorithms? Looking inside search Three key insights constrainedness “knifeedge” backbone structure 2+p-SAT Suggests branching heuristics also insight into branching mistakes 29 Inside SAT phase transition Random 3-SAT, l/n =4.3 Davis Putnam algorithm tree search through space of partial assignments unit propagation Clause to variable ratio l/n drops as we search => problems become less constrained Aside: can anyone explain simple scaling? l/n against depth/n 30 Inside SAT phase transition But (average) clause length, k also drops => problems become more constrained Which factor, l/n or k wins? Look at kappa which includes both! Aside: why is there again such simple scaling? Clause length, k against depth/n 31 Constrainedness knife-edge kappa against depth/n 32 Constrainedness knife-edge Seen in other problem domains number partitioning, … Seen on “real” problems exam timetabling (alias graph colouring) Suggests branching heuristic “get off the knife-edge as quickly as possible” minimize or maximize-kappa heuristics must take into account branching rate, max-kappa often therefore not a good move! 33 Minimize constrainedness Many existing heuristics minimize-kappa or proxies for it For instance Karmarkar-Karp heuristic for number partitioning Brelaz heuristic for graph colouring Fail-first heuristic for constraint satisfaction … Can be used to design new heuristics removing some of the “black art” 34 Backbone Variables which take fixed values in all solutions alias unit prime implicates Let fk be fraction of variables in backbone l/n < 4.3, fk vanishing (otherwise adding clause could make problem unsat) l/n > 4.3, fk > 0 discontinuity at phase boundary! 35 Backbone Search cost correlated with backbone size if fk non-zero, then can easily assign variable “wrong” value such mistakes costly if at top of search tree Backbones seen in other problems graph colouring TSP … Can we make algorithms that identify and exploit the backbone structure of a problem? 36 2+p-SAT Morph between 2-SAT and 3-SAT fraction p of 3-clauses fraction (1-p) of 2-clauses 2-SAT is polynomial (linear) phase boundary at l/n =1 but no backbone discontinuity here! 2+p-SAT maps from P to NP p>0, 2+p-SAT is NP-complete 37 2+p-SAT fk only becomes discontinuous above p=0.4 Search cost against n but NP-complete for p>0 ! search cost shifts from linear to exponential at p=0.4 recent work on backbone fragility 38 Structure Can we model structural features not found in uniform random problems? How does such structure affect our algorithms and phase transition behaviour? The real world isn’t random? Very true! Can we identify structural features common in real world problems? Consider graphs met in real world situations social networks electricity grids neural networks ... 40 Real versus Random Real graphs tend to be sparse dense random graphs contains lots of (rare?) structure Real graphs tend to have short path lengths as do random graphs Real graphs tend to be clustered L, average path length C, clustering coefficient (fraction of neighbours connected to each other, cliqueness measure) mu, proximity ratio is C/L normalized by that of random graph of same size and density unlike sparse random graphs 41 Small world graphs Sparse, clustered, short path lengths Six degrees of separation Stanley Milgram’s famous 1967 postal experiment recently revived by Watts & Strogatz shown applies to: actors database US electricity grid neural net of a worm ... 42 An example 1994 exam timetable at Edinburgh University 59 nodes, 594 edges so relatively sparse but contains 10-clique less than 10^-10 chance in a random graph assuming same size and density clique totally dominated cost to solve problem 43 Small world graphs To construct an ensemble of small world graphs morph between regular graph (like ring lattice) and random graph prob p include edge from ring lattice, 1-p from random graph real problems often contain similar structure and stochastic components? 44 Small world graphs ring lattice is clustered but has long paths random edges provide shortcuts without destroying clustering 45 Small world graphs 46 Small world graphs 47 Colouring small world graphs 48 Small world graphs Other bad news disease spreads more rapidly in a small world Good news cooperation breaks out quicker in iterated Prisoner’s dilemma 49 Other structural features It’s not just small world graphs that have been studied Large degree graphs Barbasi et al’s power-law model Ultrametric graphs Hogg’s tree based model Numbers following Benford’s Law 1 is much more common than 9 as a leading digit! prob(leading digit=i) = log(1+1/i) such clustering, makes number partitioning much easier 50 The future? What open questions remain? Where to next? Open questions Prove random 3-SAT occurs at l/n = 4.3 random 2-SAT proved to be at l/n = 1 random 3-SAT transition proved to be in range 3.003 < l/n < 4.506 random 3-SAT phase transition proved to be “sharp” 2+p-SAT heuristic argument based on replica symmetry predicts discontinuity at p=0.4 prove it exactly! 52 Open questions Impact of structure on phase transition behaviour some initial work on quasigroups (alias Latin squares/sports tournaments) morphing useful tool (e.g. small worlds, 2-d to 3-d TSP, …) Optimization v decision some initial work by Slaney & Thiebaux problems in which optimized quantity appears in control parameter and those in which it does not 53 Open questions Does phase transition behaviour give insights to help answer P=NP? it certainly identifies hard problems! problems like 2+p-SAT and ideas like backbone also show promise But problems away from phase boundary can be hard to solve over-constrained 3-SAT region has exponential resolution proofs under-constrained 3-SAT region can throw up occasional hard problems (early mistakes?) 54 Summary That’s nearly all from me! Conclusions Phase transition behaviour ubiquitous decision/optimization/... NP/PSpace/P/… random/real Phase transition behaviour gives insight into problem hardness suggests new branching heuristics ideas like the backbone help understand branching mistakes 56 Conclusions AI becoming more of an experimental science? theory and experiment complement each other well increasing use of approximate/heuristic theories to keep theory in touch with rapid experimentation Phase transition behaviour is FUN lots of nice graphs as promised and it is teaching us lots about complexity and algorithms! 57 Very partial bibliography Cheeseman, Kanefsky, Taylor, Where the really hard problem are, Proc. of IJCAI-91 Gent et al, The Constrainedness of Search, Proc. of AAAI-96 Gent & Walsh, The TSP Phase Transition, Artificial Intelligence, 88:359-358, 1996 Gent & Walsh, Analysis of Heuristics for Number Partitioning, Computational Intelligence, 14 (3), 1998 Gent & Walsh, Beyond NP: The QSAT Phase Transition, Proc. of AAAI-99 Gent et al, Morphing: combining structure and randomness, Proc. of AAAI-99 Hogg & Williams (eds), special issue of Artificial Intelligence, 88 (1-2), 1996 Mitchell, Selman, Levesque, Hard and Easy Distributions of SAT problems, Proc. of AAAI-92 Monasson et al, Determining computational complexity from characteristic ‘phase transitions’, Nature, 400, 1998 Walsh, Search in a Small World, Proc. of IJCAI-99 Watts & Strogatz, Collective dynamics of small world networks, Nature, 393, 1998 58
