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Where are the hard problems?

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Where are the hard problems? Patrick Prosser with help from Peter Cheeseman Bob Kanefsky Will Taylor APES and many more Propositional Satisfiability SAT does a truth ... – PowerPoint PPT presentation

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Title: Where are the hard problems?


1
Where are the hard problems?
  • Patrick Prosser with help from
  • Peter Cheeseman
  • Bob Kanefsky
  • Will Taylor
  • APES
  • and many more

2
Where are the hard problems?
3
Remember Graph Colouring?
Remember 3Col?
4
3 Colour me?
5
3 Colour me?
Easy?
6
3 Colour me?
7
3 Colour me?
Easy?
8
3 Colour me?
9
3 Colour me?
Easy?
10
3 Colour me?
Easy?
Does Size Matter?
11
3 Colour me?
Does size matter?
12
So, Where are the hard problems?
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Wots NP?
Nondeterministic Polynomial Problems that cannot
be solved in polynomial (P) time as far as we
know
NP-Complete (NPC) If a polytime alg can be found
for any NPC problem Then it can be adapted for
all NPC problems
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Wots SAT?
Toby?
27
Propositional Satisfiability
  • SAT
  • does a truth assignment exist that satisfies a
    propositional formula?
  • special type of constraint satisfaction problem
  • Variables are Boolean
  • Constraints are formulae
  • 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, ...
28
Wots complexity of 3SAT?
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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?

31
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
  • Whats so special about 4.3?

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  • CKT were first to report the phenomenon
  • Were they the first to see it?

39
Feldman and Golumbic 1990 Student Scheduling
Problems

Wait a minute! 1990? Real problems?
40
Gaschnig PhD thesis 1979 2nd last page
My favourite! Gaschnigs random 10 queens

41
Gaschnig 1979 Log of search effort against
constraint tightness Algorithm independent
phenomena
Rotate to view!
42
Gaschnigs Thesis, page 179
4.4.3 Cost as a Function of L A sharp Peak at L
0.6
43
  • Random CSPs ltn,m,p1,p2gt
  • n the number of variables
  • m domain size
  • p1 the probability of a constraint
  • between variables Vi and Vj
  • p2 probability Vix and Vjy are in conflict
  • lt20,10,1.0,0gt
  • easy soluble clique
  • lt20,10,1.0,1.0gt
  • easy insoluble clique
  • lt20,10,1.0,0.2gt
  • hard, phase transition, clique
  • lt20,10,0.5,0.37gt
  • Drosophilia

44
ECAI94, random csps
1994, PT for CSP, show it exists, try and locate
it (bms also at ECAI94) And lunch with Barbara,
Toby, and Ian

45
Frost and Dechter AAAI94

1994 again, Frost and Dechter tabulate, use this
for comparison of algs (CKTs first goal!)
46
Bessiere AIJ65 1994
1994 again! A problem in P

47
Constrainedness
ltSolgt is expected number of solutions N is
log_2 of the size of the state space
k 0, all states are solutions, easy,
underconstrained
k
, ltSolgt is zero, easy, overconstrained
k 1, critically constrained, 50 solubility,
hard
Applied to CSP, TSP, 3-SAT, 3-COL, Partition,
HC, ?
48
  • 1994
  • critical ratio of clauses to variables in 3SAT
  • 1995
  • applied techniques from statistical mechanics to
    analysis
  • 1996
  • Kappa, a theory of constrainedness
  • applies in CSP, 3-SAT NumPart, TSP!, ...
  • kappa based heuristics
  • P/NP phase transition (2p)-SAT
  • At p 0.4

49
  • 1997
  • Kappa holds in P, achieving arc-consistency
  • Empirically derive complexity of AC3
  • Derive existing heuristics for revision ordering
    in AC3
  • 1998
  • Expectation of better understanding of behaviour
    of algorithms and heuristic
  • What happens inside search?

50
  • 1999
  • Kappa for QSAT
  • 2000
  • the backbone
  • 2001
  • backbone heuristics
  • 2000 and beyond
  • Physics takes over?

51
  • Conclusion?
  • More to it than just P and NP
  • we are now learning about the structure of
    problems
  • the behaviour of algorithms
  • using this to solve the problems!
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