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

Transcription

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

2 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 2

3 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 3

4 10 variable 3-SAT example each clause gives a nogood inconsistent assignment for the 3 variables e.g., v 1 =true, v 4 =false, v 5 =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 4

5 search tree examined states v 1 =true v 1 =false other consistent states 1st solution another solution 34 nogoods 5 many solutions little pruning low cost

6 search tree 55 nogoods one solution some pruning 6 high cost

7 search tree 110 nogoods no solutions much pruning 7 low cost

8 search cost cost underconstrained nogoods no solutions overconstrained 8

9 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 9

10 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 10

11 outline statistical approach to search local properties and global behavior problem ensembles phase transitions impact of structure summary 11

12 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 12

13 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 13

14 problem ensembles class of problem instances probability for each instance why? focus on typical behaviors in class (averagecase) not single instance, not worst-case 14

15 parameterized ensembles problem size properties of constraints e.g., number, tightness can also specify global properties e.g., number of solutions 15

16 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 16

17 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 17

18 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 18

19 local measures: global behavior local measure of problem structure number and tightness of constraints global behaviors probability for a solution search cost solution quality 19

20 outline statistical approach to search phase transitions decision problems optimization other complexity classes impact of structure summary 20

21 phase transitions abrupt changes in global property of typical instance in a problem ensemble characteristic of problem ensemble, not problem instance 21

22 phase transition phenomena physics: magnets, superconductors vs. temperature geology: fluid percolation vs. porosity of rocks biology epidemic spread vs. population density 22

23 phase transition: boiling small change in average energy per molecule (temperature) gives large change in overall state 23

24 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 24

25 phase transitions in AI pruning in heuristic search associative memory models pattern matching 25

26 outline statistical approach to search phase transitions decision problems optimization other complexity classes impact of structure summary 26

27 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,..,2 L ] 27

28 observe regularity search cost varies greatly among problem instances of same size hard instances rare and concentrated hard instance 28

29 intuition few constraints: many solutions many constraints: no solutions bad choices pruned quickly intermediate number of constraints: few solutions bad choices pruned only after many steps EASY HARD 29

30 typical cost of search heuristics hard cases: near phase transitions fastest exponential growth of cost easy cases: underconstrained: many solutions overconstrained: rapid pruning large size behavior of many NP problem ensembles search methods (but not generate & test) typical cost small size underconstrained overconstrained 30 number of constraints

31 typical behavior of solubility for large problem ensembles fraction of soluble instances drops suddenly from near 1 to near 0 close to location of <S>=1 prob(soln) 1 0 few soluble problems but with very many solutions <S> exp. large # constraints 31 <S> =1 <S> exp. small

32 solubility transition and cost high costs associated with drop in solutions not the whole story: cost peak for problems with fixed number of solutions! 32

33 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 33

34 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 34

35 expected cost d=20 d=10 d=4 cost constant exponential z 35

36 expected cost average branching ratio: z = b p avg. cost from depth j: C j =1+z C j+1 avg. cost from root d z j = 1 zd +1 j =0 1 z smooth fn. of z for finite d has abrupt changes when d infinite z z < 1 d z = 1 z d+1 z 1 z > 1

37 hard graph coloring many consistent colorings for much of the graph few for the whole graph hence much backtracking 37

38 relate cost to structure color k of n nodes, with 3 colors 3 k possible colorings most with about equal use of each color random edge in subgraph with prob. edge in subgraph links same colors with (k/n) 2 prob. 1/3 38

39 partial colorings assuming independent constraints, expected number of partial colorings of k nodes is N = 3 k 1 1 k 3 k n 2 e 39

40 behavior few edges: monotonic increase, little backtrack more edges: hard to continue beyond maximum, much backtrack many edges: no solutions, pruning terminates search early 40

41 behavior: 100 nodes ln(# consistent colorings) e=75 e=200 e=300 # colored nodes edges prune large subgraphs more than small ones 41

42 independence assumption simplifies analysis gives qualitative behavior reasonable quantitative accuracy cf., mean-field physics theories 42

43 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 43

44 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 44

45 Computational Complexity and Phase Transitions of Optimization Problems 45

46 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? 46

47 Combinatorial Optimization: Example 2 Interleaving decision and execution: Decision making under limited information and restricted resources Collecting information with restricted computational resources Exploration by lookahead search Which path to take? decision 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) 47

48 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. 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

49 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 49

50 Abstract Model for Complexity Analysis Incremental random tree Node = states Edges = state transitions Branching factor b= # of children Tree depth d 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 Goal state 50

51 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 51

52 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!? 52

53 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 eb C+ e1 C+ e2 C+ eb 53

54 Complexity Analysis: The Main Results Optimal solution cost C* (as d ) bp<1 C*/d bp=1 C*/loglog d 1 bp>1 C* bounded by a constant Expected complexity (# of nodes generated as d ) algorithm bp<1 bp=1 bp>1 Best-first search Depth-first branch and bound ( d ) optimal ( d ) asymptotic optimal 2 ( d ) optimal ( d ) optimal 3 2 O( d ) O( d ) Node costs can be discrete or continuous 54

55 Complexity Analysis: History J. Pearl ( ) 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 ( ) Complexity of linear-space search algorithms (DFBnB, IDA*, RBFS) on incremental random tree 55

56 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 56

57 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) 57

58 Search Anomaly of Sliding-tile Puzzle Revised Control parameter bp = expected # of same-cost children 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 (Zhang&Korf, 1995) 58

59 Phase Transitions in the ATSP Question: Does the ATSP with distances from {1,2,,r} have the same average-case complexity as r increases? Anwser: No, due to phase transitions (Zhang&Korf, 1996) Assignment Problem (AP) is the cost function for the ATSP, which can be computed in O(n^3) time. 59

60 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 60

61 Phase Transitions Decision vs. Optimization Decision problem Finding an YES/NO answer Easy-hard-easy phase transitions Optimization problem Finding an optimal solution Easy-hardphase transitions 61

62 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) 62

63 Decision vs. Optimization Quality/Complexity Tradeoff (Experiments) New, shifted phase transitions in MAX-3SAT (25 variables) (zhang, 2001) 63

64 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 64

65 Backbone: Behavior of Finding All Solutions Why finding an optimal solution is hard when a problem is overconstrained? (zhang, 2001) 65

66 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! 66

67 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! 67

68 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. 68

69 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 69

70 Exploiting Phase Transitions What do we do with a difficult problem in the exponential region? 70

71 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) 71

72 Exploiting Phase Transitions: Parametric Transformation The Method Increasing the number of zero-cost edges in search space 72

73 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 73

74 Exploiting Phase Transitions: Parametric Transformation Application 74

75 Exploiting Phase Transitions: Structural Transformation Idea and Method Prune large cost edges in search space (Zhang, 1998, 2001) 75

76 Exploiting Phase Transitions: Structural Transformation Applications 76

77 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 77

78 Beyond NP Other complexity classes Phase transitions in P, PSPACE, Structure Backbones, 2+p-SAT, small world topology, Heuristics Constrainedness knife-edge, minimize constrainedness,... 78

79 Before we begin A little history...

80 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! 80

81 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.

82 Random 2-SAT 2-SAT is P linear time algorithm 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 82 x1 v x2, -x2 v x3, -x1 v x3,

83 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 83

84 Phase transitions above NP PSpace QSAT (SAT of QBF) 84

85 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] 85

86 Exact phase boundaries in NP Random 3-SAT is only known within bounds 3.26 < l/n < Recent result gives an exact NP phase boundary Are there any NP phase boundaries known exactly? 1-in-k SAT at l/n = 2/k(k-1) 2nd order transition (like 2- SAT and unlike 3-SAT) 1st order transitions not a characteristic of NP as has been conjectured 86

87 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?

88 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

89 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? 89

90 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 90

91 2+p-SAT phase transition 91

92 2+p-SAT phase transition l/n p 92

93 2+p-SAT phase transition Lower bound are the 2-clauses (on their own) UNSAT? n.b. 2-clauses are much more constraining than 3- clauses p <= 0.4 transition occurs at lower bound 3-clauses are not contributing! 93

94 2+p-SAT backbone fk becomes discontinuous for p>0.4 but NP-complete for p>0! 6HDUFKFRVWDJDLQVWQ search cost shifts from linear to exponential at p=0.4 similar behavior seen with local search algorithms 94

95 Structure How do we model structural features found in real problems? How does such structure affect phase transition behaviour?

96 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

97 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 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 97

98 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... 98

99 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 99

100 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? 100

101 Small world graphs ring lattice is clustered but has long paths random edges provide shortcuts without destroying clustering 101

102 Small world graphs 102

103 Small world graphs 103

104 Colouring small world graphs 104

105 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

106 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 106

107 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] 107

108 QCP phase transition 108

109 QCP phase transition 109

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

111 Looking inside search Constrainedness knife-edge problems are critically constrained between SAT and UNSAT Suggests branching heuristics also insight into branching mistakes 111

112 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? 112 OQDJDLQVWGHSWKQ

113 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 113

114 Constrainedness knife-edge NDSSDDJDLQVWGHSWKQ 114

115 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! 115

116 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 116