Phase Transitions and Structure in Combinatorial ... - Semantic Scholar

Phase Transitions and Structure in Combinatorial Problems Carla P. Gomes Tad Hogg Toby Walsh Weixiong Zhang

a statistical regularity in search • relates structure to behavior – e.g., problem features => search cost

• analogous to physical phase transitions • useful to improve search heuristics – i.e., new problem solving strategies

Gomes, Hogg, Walsh, Zhang

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IJCAI-01 Tutorial on Phase Transitions and Structures-SA4

combinatorial search e.g.: scheduling, circuit design, cryptography, theorem proving, games, matching models to data, genetics, ...

• types – decision (NP): rapid test of candidate solutions – optimization

• behavior – worst case (e.g., is P=NP?) – typical performance of heuristics


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10 variable 3-SAT example • each clause gives a “nogood” – inconsistent assignment for the 3 variables • e.g., v1=true, v4=false, v5=false

• search method: depth-1st backtrack – cost: steps to find a solution (if any)

• examine search tree as nogoods added – i.e., a particular sequence of clauses, randomly generated


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search tree v1=true

v1=false

examined states

34 nogoods 1st solution Gomes, Hogg, Walsh, Zhang

another solution

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other consistent states

many solutions little pruning IJCAI-01 lowTutorial coston Phase Transitions and Structures-SA4

search tree

55 nogoods Gomes, Hogg, Walsh, Zhang

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one solution some pruning IJCAI-01 highTutorial coston Phase Transitions and Structures-SA4

search tree

110 nogoods Gomes, Hogg, Walsh, Zhang

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no solutions much pruning IJCAI-01 lowTutorial coston Phase Transitions and Structures-SA4

search cost 140 120

cost

100

no solutions

80 60 40 20 0 0

25

underconstrained Gomes, Hogg, Walsh, Zhang

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75 nogoods 8

100

125

150

overconstrained IJCAI-01 Tutorial on Phase Transitions and Structures-SA4

observations • peak in cost vs. number of nogoods – competition between • decreasing number of solutions • increasing search path pruning

• peak corresponds to loss of last solution • peak location depends on choice of nogoods


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generality • example uses – one search method – one sequence of problem instances

• Why is this interesting? – similar behavior for many • search problems • search techniques

– hence: simple parameter indicates likely cost


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outline • statistical approach to search – local properties and global behavior

– problem ensembles • phase transitions • impact of structure • summary


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statistical approach to search • How do many repeated local decisions combine to give global behavior? – e.g., heuristic choice accuracy => search cost

• key issue: – local properties => global behavior – addressed by statistical mechanics


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statistical approach • identify problem ensemble • identify structure parameters – easy to measure (computationally efficient) – describe an average, local property of problem

• relate parameters to global behavior – e.g., search cost


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problem ensembles • class of problem instances • probability for each instance • why? – focus on typical behaviors in class (averagecase) – not single instance, not worst-case


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parameterized ensembles • problem size • properties of constraints – e.g., number, tightness

• can also specify global properties – e.g., number of solutions


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simple ensemble examples • random k-SAT: n variables, m clauses – each clause picked independently • allowing duplicates

• random graphs: n nodes, e edges – each edge picked randomly • but no duplicates


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“real” problems • constraints tend to be local and more clustered than random – e.g., small worlds graphs

• harder to work with than easily generated random ensembles – both analytically and empirically


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nature of constraints • physical interactions – strength decreases with distance

• designed artifacts – often nearly decomposable, hierarchical

• evolved systems – e.g., ecology, economy, common law, social networks


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local measures: global behavior • local measure of problem structure – number and tightness of constraints

• global behaviors – probability for a solution – search cost – solution quality


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outline • statistical approach to search • phase transitions – decision problems – optimization – other complexity classes • impact of structure • summary


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phase transitions • abrupt changes in global property of typical instance in a problem ensemble • characteristic of problem ensemble, not problem instance


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phase transition phenomena • physics: – magnets, superconductors vs. temperature

• geology: – fluid percolation vs. porosity of rocks

• biology – epidemic spread vs. population density


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phase transition: boiling • small change in average energy per molecule (temperature) • gives large change in overall state


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commonality? • key idea: abrupt change in overall system property when size is large • why a useful general concept? – – – –

mathematical commonality in spite of different details suggests analyses, e.g., finite-size scaling universality: insight from simple models with “essential features”


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phase transitions in AI • pruning in heuristic search • associative memory models • pattern matching


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outline • statistical approach to search • phase transitions – decision problems – optimization – other complexity classes • impact of structure • summary


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some decision problems • SAT – random 3-SAT: clause/variables near 4.2

• graph coloring – random graph, 3-COL: edges/nodes near 2.2

• number partitioning – L integers, random in [0,..,2L]


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observe regularity • search cost varies greatly among problem instances of same size • hard instances rare and concentrated hard instance


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intuition • few constraints: many solutions • many constraints: no solutions EASY

– bad choices pruned quickly

• intermediate number of constraints: – few solutions HARD – bad choices pruned only after many steps


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typical cost of search heuristics hard cases: • near phase transitions • fastest exponential growth of cost

large size

behavior of many • NP problem ensembles • search methods (but not generate & test)

easy cases: • underconstrained: many solutions • overconstrained: rapid pruning typical cost

small size

underconstrained Gomes, Hogg, Walsh, Zhang

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overconstrained

IJCAI-01 Tutorial on Phase number of constraints Transitions and Structures-SA4

typical behavior of solubility • for large problem ensembles – fraction of soluble instances drops suddenly from near 1 to near 0 – close to location of =1 =1

prob(soln)

1

few soluble problems but with very many solutions

0

exp. large

# constraints Gomes, Hogg, Walsh, Zhang

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exp. small IJCAI-01 Tutorial on Phase Transitions and Structures-SA4

solubility transition and cost • high costs associated with drop in solutions • not the whole story: cost peak for problems with fixed number of solutions !


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theory • Why the association of high cost with transition, on average? • simplified models of the behavior: – exact: simple tree search • trivial problem • simple illustration of abrupt change

– approximate: graph coloring


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tree search • branching ratio b, depth d • no solution – e.g., prior choices are already inconsistent

• heuristic prunes with prob. p – average branching ratio: z = b p

• expected cost for depth-1st backtrack


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expected cost d=10

d=4

cost

d=20

constant

exponential

z


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expected cost • average branching ratio: z = b p • avg. cost from depth j: Cj=1+z Cj+1 • avg. cost from root

 1 z 1  has abrupt changes when d infinite z − 1 36 IJCAI-01 Tutorial on Phase Gomes, Hogg, Walsh, Zhang Transitions and Structures-SA4

hard graph coloring • many consistent colorings for much of the graph • few for the whole graph • hence much backtracking


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relate cost to structure • color k of n nodes, with 3 colors • 3k possible colorings – most with about equal use of each color

• random edge in subgraph with prob. 2 • edge in subgraph links same colors with ≈ (k/n) prob. 1/3


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partial colorings • assuming independent constraints, expected number of partial colorings of k nodes is

 1 k k Nk = 3  1−    3  n Gomes, Hogg, Walsh, Zhang

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2

  

e


behavior • few edges: monotonic increase, little backtrack • more edges: hard to continue beyond maximum, much backtrack • many edges: no solutions, pruning terminates search early


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ln(# consistent colorings)

behavior: 100 nodes e=75

e=200

e=300

# colored nodes edges prune large subgraphs more than small ones Gomes, Hogg, Walsh, Zhang

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independence assumption • • • •

simplifies analysis gives qualitative behavior reasonable quantitative accuracy cf., “mean-field” physics theories


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which ensemble? • examples – random graphs – small-world graphs

• cf., physical systems – all states consistent with (few) parameters are equally likely – accuracy is an empirical question – e.g., classical => quantum statistical mechanics


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open questions • math: prove thresholds in simple ensembles • empirical: which ensembles are realistic? – perhaps additional parameters required – e.g., random graph vs. small worlds graph


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Computational Complexity and Phase Transitions of Optimization Problems


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Combinatorial Optimization: Example 1 •

The Traveling Salesman Problem (TSP) – Given: a set of cities and inter-city distances – Goal: a shortest tour visit each city once – Asymmetric TSP (ATSP): distance(i,j) distance(j,i) – Many real-world applications • VLSI routing • Scheduling

– A well-studied NP-hard problem – TSP with distances 1 and 2 is still NP-hard! (Papadimitriou&Yannakakis 93) – NP-hardness is a worst-case measure – How does the average-case complexity behave?


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Combinatorial Optimization: Example 2 • Interleaving decision and execution: Decision making under limited information and restricted resources

Which path to take? decision

– Collecting information with restricted computational resources • Exploration by lookahead search

– Reasoning based on limited information • Calculation

Lookahead to collect information

– Executing the best decision • Action

• This idea is the foundation of almost all game playing programs (Shannon 1950) Gomes, Hogg, Walsh, Zhang

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Combinatorial Optimization: Example 2 •

Lookahead search in sliding-tile puzzle: • backup the minimal cost at a fixed depth

•

f=g+h: g=depth of a node, h= Manhattan distance to the goal. In one step, g increases by one; h increases or decreases by one; and so f increases by 2 or remains the same. 1 2 3 4 5 6 7 8 1 2 34 5 67 8 1 2 3 45 6 78

1 4 2 3 5 6 7 8


Anomaly of sliding-tile puzzles: For a given search depth, a large puzzle is easier to search; or for a given amount of computation, a large puzzle can be searched to a deeper level.

1 2 3 8 4 7 6 5 3 1 2 4 5 6 7 8

3 1 2 4 5 6 7 8

3 1 2 6 4 5 7 8

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Content • Expected complexity of optimization search – Abstract (statistical) tree model – Complexity results

• Phase transitions of optimization search • Relationship between phase transitions in decision and optimization problems • Backbone phase transitions • New search strategies by exploiting phase transitions Gomes, Hogg, Walsh, Zhang

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Abstract Model for Complexity Analysis • Incremental random tree – Node = states

– Edges = state transitions

– Tree depth d

7

– Edge costs (operator costs): i.id non-negative random variable (may take 0) – Node costs (state quality): sum of edge costs from the root to the nodes 50

2

– Branching factor b= # of children


0

1

2

5

2

3

4

5

6

8

3

7

4

Goal state


Complexity Analysis: A Quiz • • •

Given two random trees generated with the same edge cost distribution Finding the goal nodes (minimum-cost leaf nodes) using the same search algorithm (BFS, DFBnB, IDA*, …) Which tree is easier to search?

• Random tree 1: b=3, d=100


• Random tree 1: b=2, d=100

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Complexity Analysis: Anomaly • random tree with edge costs uniformly chosen from {0,1,2,3,4} • A tree with a larger branching factor is more difficult to search than one with a smaller branching factor !?


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Complexity Analysis: Control Parameter • Control parameter: – b = the mean branching factor – p = probability that an edge has cost 0 (edge costs can be discrete or continuous) – bp = expected number of children having the same cost as the parent

• A local property determines a global behavior! C e1

e2

C+ e1 C+ e2 Gomes, Hogg, Walsh, Zhang

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eb C+ eb IJCAI-01 Tutorial on Phase Transitions and Structures-SA4

Complexity Analysis: The Main Results )

• Optimal solution cost C* (as d bp1 1

C* bounded by a constant

• Expected complexity (# of nodes generated as d algorithm

bp1

Best-first search

( d) optimal

( d2 ) optimal

(d) optimal

Depth-first branch and bound

( d) asymptotic optimal

O( d 3 )

O( d 2 )

)

Node costs can be discrete or continuous Gomes, Hogg, Walsh, Zhang

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Complexity Analysis: History • J. Pearl (1980-1984) Analysis of A* (BFS) • R. Karp & J. Pearl (1983) Complexity of BFS on incremental random binary tree with {0,1} edge costs • C.J.H. McDiarmid (1990) Complexity of BFS on incremental random tree • W. Zhang & R.E. Korf (1992-1995) Complexity of linear-space search algorithms (DFBnB, IDA*, RBFS) on incremental random tree


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Content • Expected complexity of optimization search • Phase transitions of optimization search – Phase Transitions and Phase diagram – Answers to anomaly – Phase transitions in the TSP

• Relationship between phase transitions in decision and optimization problems • Backbone phase transitions • New search strategies by exploiting phase transitions Gomes, Hogg, Walsh, Zhang

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Phase Transition and Phase Diagram • Control parameter bp = expected number of children having the same cost as the parent

• Complexity: exponential vs. polynomial

(Zhang&Korf, 1995)


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Search Anomaly of Sliding-tile Puzzle Revised • Control parameter bp = expected # of same-cost children

(Zhang&Korf, 1995)

• f=g+h • g=depth of a node, which increases by one for each move • h=Manhattan distance, which increases or decreases by one for each move with probability ½ approximately (a tile moved toward or away from its goal position)

• Edge costs in the search tree take values 0 or 2; p=prob(cost zero)§1/2. • b increases with puzzle size • bp increases with puzzle size Gomes, Hogg, Walsh, Zhang

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Phase Transitions in the ATSP • Question: Does the ATSP with distances from {1,2,…,r} have the same average-case complexity as r increases? (Zhang&Korf, 1996) • Anwser: No, due to phase transitions

•

Assignment Problem (AP) is the cost function for the ATSP, which can be computed in O(n^3) time.


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Content • Expected complexity of optimization search • Phase transitions of optimization search • Relationship between phase transitions in decision and optimization problems • Backbone phase transitions • New search strategies by exploiting phase transitions


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Phase Transitions – Decision vs. Optimization • Decision problem

• Optimization problem

– Finding an YES/NO answer – Easy-hard-easy phase transitions


– Finding an optimal solution – Easy-hard phase transitions

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Phase Transitions –

Decision versus Optimization (A Closer Look) • Different phase transition patterns – Decision problem has easy-hard-easy transition pattern – Optimization problem has easy-hard transition pattern

• Complexity discrepancy - Tighter constraints have different impact – They make a decision problem easier – They make an optimization problem more difficult

• Optimization is hard! (Zhang, 2001, see also Slaney&Walsh, 2001, later slides) Gomes, Hogg, Walsh, Zhang

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Decision vs. Optimization Quality/Complexity Tradeoff (Experiments) • New, shifted phase transitions in MAX-3SAT (25 variables) (zhang, 2001)


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Content • Expected complexity of optimization search • Phase transitions of optimization search • Relationship between phase transitions in decision and optimization problems • Backbone phase transitions • New search strategies by exploiting phase transitions


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Backbone: Behavior of Finding All Solutions Why finding an optimal solution is hard when a problem is overconstrained?

(zhang, 2001)


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Backbone: Behavior of Finding All Solutions • Number of solutions of MAX-3SAT (25 variables): – The rate that the number of solutions changes is different in overconstrained and underconstrained regions!


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Backbone: Behavior of Finding All Solutions • Number of clauses unsatisfied in and complexity of MAX-3SAT (25 variables): – They all have different patterns in the overconstrained and underconstrained regions!


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Backbone: Behavior of Finding All Solutions • Backbone: The set of variables that have fixed values in all solutions • There is a phase transition in backbone in MAX 3-SAT (25 variables) • Backbone and satisfiability is roughly linearly correlated.


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Content • Expected complexity of optimization search • Phase transitions of optimization search • Relationship between phase transitions in decision and optimization problems • Backbone phase transitions • New search strategies by exploiting phase transitions – Parametric transformations – Structural transformations


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Exploiting Phase Transitions • What do we do with a difficult problem in the exponential region?


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Exploiting Phase Transitions: Parametric Transformation – Basic Idea • Treat a difficult problem in the exponential region as if it is an easy one in the polynomial region (Zhang&Pemberton, 94, Pemberton&Zhang, 96)


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Exploiting Phase Transitions: Parametric Transformation – The Method • Increasing the number of “zero-cost edges” in search space


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Exploiting Phase Transitions: Parametric Transformation – Improvement • For a real-world problem, edge costs are not known in advance, but can be learned on the fly. • If a solution found is not good enough, another transformation is applied with a smaller epsilon value – iterative parametric transformation


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Exploiting Phase Transitions: Parametric Transformation – Application


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Exploiting Phase Transitions: Structural Transformation – Idea and Method • Prune large cost edges in search space (Zhang, 1998, 2001)


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Exploiting Phase Transitions: Structural Transformation – Applications


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Take Home Message (Summary) • Finding optimal solution is hard than deciding solubility • Phase transitions can be used to characterize complex problems and their behavior • Understanding phase transitions can help to design and develop more efficient search algorithms


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Beyond NP Other complexity classes Phase transitions in P, PSPACE, …

Structure Backbones, 2+p-SAT, small world topology, …

Heuristics Constrainedness knife-edge, minimize constrainedness, ... Gomes, Hogg, Walsh, Zhang

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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!


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Other complexity classes Enough of the history, are phase transitions just in NP? Conjecture in Cheeseman et al paper that phase transitions distinguish P from NP.

Random 2-SAT • 2-SAT is P – linear time algorithm x1 v x2, -x2 v x3, -x1 v x3, …

• Random 2-SAT displays “classic” phase transition – l/n < 1, almost surely SAT – l/n > 1, almost surely UNSAT – complexity peaks around l/n=1 Gomes, Hogg, Walsh, Zhang

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Phase transitions in P • 2-SAT – l/n=1

• Horn SAT – transition not “sharp”

• Arc-consistency – rapid transition in whether problem can be made AC – peak in (median) checks Gomes, Hogg, Walsh, Zhang

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Phase transitions above NP • PSpace – QSAT (SAT of QBF)


84


Phase transitions above NP • PSpace-complete – QSAT (SAT of QBF) – stochastic SAT – modal SAT

• PP-complete – polynomial-time probabilistic Turing machines – counting problems – #SAT(>= 2^n/2) [Bailey, Dalmau, Kolaitis IJCAI-2001] Gomes, Hogg, Walsh, Zhang

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Exact phase boundaries in NP • Random 3-SAT is only known within bounds – 3.26 < l/n < 4.596

Are there any NP phase boundaries known exactly?

• Recent result gives an exact NP phase boundary – 1-in-k SAT at l/n = 2/k(k-1) – 2nd order transition (like 2SAT and unlike 3-SAT)


1st order transitions not a characteristic of NP as has been conjectured

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Structure Can we identify structure in (random) problems that makes problems hard? How do we model structural features found in real problems? How does such structure affect phase transition behaviour?

Backbone • Variables which take fixed values in all solutions – alias unit prime implicates

• Let fk be fraction of variables in backbone – in random 3-SAT l/n < 4.3, fk vanishing (otherwise adding clause could make problem unsat) l/n > 4.3, fk > 0

discontinuity at phase boundary!


88


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 – coloring, TSP, blocks world planning … see [Slaney, Walsh IJCAI-2001]

Can we make algorithms that identify and exploit the backbone structure of a problem?


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2+p-SAT • Morph between 2-SAT and 3SAT – 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


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2+p-SAT phase transition


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2+p-SAT phase transition

l/n

p Gomes, Hogg, Walsh, Zhang

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2+p-SAT phase transition • Lower bound – are the 2-clauses (on their own) UNSAT? – n.b. 2-clauses are much more constraining than 3clauses

• p 0.4

6HDUFKFRVWDJDLQVWQ

– but NP-complete for p>0 !

• search cost shifts from linear to exponential at p=0.4 • similar behavior seen with local search algorithms


94


Structure How do we model structural features found in real problems? How does such structure affect 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 ...


96


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

L, average path length C, clustering coefficient (fraction of neighbours connected to each other, cliqueness measure)

– as do random graphs

• Real graphs tend to be clustered

mu, proximity ratio is C/L normalized by that of random graph of same size and density

– unlike sparse random graphs Gomes, Hogg, Walsh, Zhang

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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: • • • • Gomes, Hogg, Walsh, Zhang

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actors database US electricity grid neural net of a worm ... IJCAI-01 Tutorial on Phase Transitions and Structures-SA4

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


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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? Gomes, Hogg, Walsh, Zhang

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Small world graphs

• ring lattice is clustered but has long paths • random edges provide shortcuts without destroying clustering Gomes, Hogg, Walsh, Zhang

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Small world graphs


102


Small world graphs


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Colouring small world graphs


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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


105


Other structural features It’s not just small world graphs that have been studied

• Large degree graphs – Barbasi et al’s power-law model [Walsh, IJCAI 2001]

• 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 Gomes, Hogg, Walsh, Zhang

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Another structured problem • Quasigroup completion problem (QCP) Can we complete a partial Latin square?

"

• Regular structure found in real problems – sports tournaments – fibre optic routing See [Kautz et al, IJCAI 2001]


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QCP phase transition


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QCP phase transition


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Heuristics What do we understand about problem hardness at the phase boundary? How can this help build better heuristics?

Looking inside search • Constrainedness “knife-edge” – problems are critically constrained between SAT and UNSAT

• Suggests branching heuristics – also insight into branching mistakes


111


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? Gomes, Hogg, Walsh, Zhang

OQDJDLQVWGHSWKQ 112


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? &ODXVHOHQJWKNDJDLQVWGHSWKQ Gomes, Hogg, Walsh, Zhang

113


Constrainedness knife-edge NDSSDDJDLQVWGHSWKQ


114


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! Gomes, Hogg, Walsh, Zhang

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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”


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