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## Phase transition behaviour

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### Before we begin

### Phase transitions

### Locating phase transitions

### Algorithms at the phase boundary

### Structure

### Summary

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?

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

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!

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
- 3-SAT
- formulae in clausal form with 3 literals per clause
- remains NP-complete

(x1 v x2) & (-x2 v x3 v -x4)

x1/ True, x2/ False, ...

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?

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?

Random 3-SAT

- Complexity peak coincides with solubility transition
- l/n < 4.3 problems under-constrained and SAT
- l/n > 4.3 problems over-constrained and UNSAT
- l/n=4.3, problems on “knife-edge” between SAT and UNSAT

“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!

“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
- ...

“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)
- ...

“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
- ...

“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)

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, …)

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

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!

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, under-constrained
- kappa=infty, <Sol>=0, over-constrained
- kappa=1, <Sol>=1, critically constrained

phase boundary

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

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

Number partition phase transition

Prob(perfect partition) against kappa

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

Rescaled phase transition

Prob(perfect partition) against gamma

Rescaled search cost

Optimization cost against gamma

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?

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 “knife-edge”
- backbone structure
- 2+p-SAT
- Suggests branching heuristics
- also insight into branching mistakes

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

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

Constrainedness knife-edge

kappa against depth/n

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!

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”

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!

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?

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

2+p-SAT

- fk only becomes discontinuous above p=0.4
- but NP-complete for p>0 !
- search cost shifts from linear to exponential at p=0.4
- recent work on backbone fragility

Search cost against n

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
- ...

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

unlike sparse random graphs

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

Real versus RandomSmall 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
- ...

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

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?

Small world graphs

- ring lattice is clustered but has long paths
- random edges provide shortcuts without destroying clustering

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

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

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!

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

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?)

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

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!

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

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