Post on 20-Jun-2015
description
Pancake Flipping Is Hard
Laurent BULTEAU, Guillaume FERTIN, Irena RUSULINA, Université de Nantes
Feb. 21st, 2012
Pancake Flipping Problem
L. Bulteau Pancake Flipping Is Hard 2/28
Pancakes
we are given a stack of pancakes, all of different sizes
we want to rearrange it into a beautiful pyramidal stackwe have a spatula, to flip the top of the stackwe are lazy
L. Bulteau Pancake Flipping Is Hard 3/28
Pancakes
we are given a stack of pancakes, all of different sizeswe want to rearrange it into a beautiful pyramidal stack
we have a spatula, to flip the top of the stackwe are lazy
L. Bulteau Pancake Flipping Is Hard 3/28
Pancakes
we are given a stack of pancakes, all of different sizeswe want to rearrange it into a beautiful pyramidal stackwe have a spatula, to flip the top of the stack
we are lazy
L. Bulteau Pancake Flipping Is Hard 3/28
Pancakes
we are given a stack of pancakes, all of different sizeswe want to rearrange it into a beautiful pyramidal stackwe have a spatula, to flip the top of the stackwe are lazy
L. Bulteau Pancake Flipping Is Hard 3/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
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PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
PancakesExample
L. Bulteau Pancake Flipping Is Hard 4/28
The problem to be solved
ProblemGiven a stack of n pancakes, how can it be arranged with as littleeffort as possible?
L. Bulteau Pancake Flipping Is Hard 5/28
Other points of view
“Pancake view”Given a stack of n pancakes, how can it be arranged with as littleeffort as possible?
Formal problem : MIN-SBPR
Given a permutation π of {1, . . . , n}, compute the prefix reversaldistance between π and the Identity, written prd(π).
“Biology view”
Given two genomes using the same n genes, how many steps havebeen used in evolution between one and the other?
L. Bulteau Pancake Flipping Is Hard 6/28
Other points of view
Pancakes Formal BiologyPancake Integer GeneStack Permutation Genome
Nice stack Identity Reference genomeFlip Prefix reversal Evolution step
We are lazy Minimization formulation Parsimony principle(distance)
L. Bulteau Pancake Flipping Is Hard 7/28
Other points of view
Pancakes Formal BiologyPancake Integer GeneStack Permutation Genome
Nice stack Identity Reference genomeFlip Prefix reversal Evolution step
We are lazy Minimization formulation Parsimony principle(distance)
L. Bulteau Pancake Flipping Is Hard 7/28
Other points of view
Pancakes Formal BiologyPancake Integer GeneStack Permutation Genome
Nice stack Identity Reference genomeFlip Prefix reversal Evolution step
We are lazy Minimization formulation Parsimony principle(distance)
L. Bulteau Pancake Flipping Is Hard 7/28
Pancake Problem
Complexity: NP-complete
Related results:Reversal distance, not necessarily prefix:
NP-complete (APX-hard) for unsigned permutations,polynomial for signed permutations.
Burnt pancakes variant, or Prefix Reversal Distance for signedpermutations:
complexity unknown.Algorithms:
polynomial-time algorithm for a subclass of signedpermutations (simple permutations [Labarre, Cibulka, 2011])2-approximation algorithm
L. Bulteau Pancake Flipping Is Hard 8/28
Known bounds
Upper bound
prd(π) ≤ 2(n − 1)
Repeat at most n − 1 timesFind the largest unsorted pancake, flip it to the topFlip it back to its destination
At most 2(n − 1) flips to sort a stack.
L. Bulteau Pancake Flipping Is Hard 9/28
Known bounds
Upper bound
prd(π) ≤ 2(n − 1)Anything better ?
L. Bulteau Pancake Flipping Is Hard 9/28
Known bounds
Upper bound
prd(π) ≤ 2(n − 1)Anything better ? – Yes :prd(π) ≤ (5n + 5)/3 [Gates & Papadimitriou, 1979]
L. Bulteau Pancake Flipping Is Hard 9/28
Known bounds
Upper bound
prd(π) ≤ 2(n − 1)Anything better ? – Yes :prd(π) ≤ (5n + 5)/3 [Gates & Papadimitriou, 1979]prd(π) ≤ (18n)/11 + O(1) [Chitturi et al., 2009]
L. Bulteau Pancake Flipping Is Hard 9/28
Known bounds
Upper bound
prd(π) ≤ (18n)/11 + O(1) [Chitturi et al., 2009]
Lower boundprd(π) ≥ db(π)
Breakpoint at position i if:i < n and π(i + 1) 6= π(i)± 1i = n and π(n) 6= n
db(π) : number of breakpoints
At most one breakpoint is removed with each flip
L. Bulteau Pancake Flipping Is Hard 9/28
Known bounds
Upper bound
prd(π) ≤ (18n)/11 + O(1) [Chitturi et al., 2009]
Lower boundprd(π) ≥ db(π)
Breakpoint at position i if:i < n and π(i + 1) 6= π(i)± 1i = n and π(n) 6= n
db(π) : number of breakpointsAt most one breakpoint is removed with each flip
L. Bulteau Pancake Flipping Is Hard 9/28
Our result
Reduction from 3-SAT: from a formula φ, create apermutation πφ such that prd(πφ) = db(πφ) iff φ is satisfiable.Given a permutation π, deciding wether π can be sorted withno more than db(π) flips is NP-hard.
MIN-SBPR is NP-hard (hence NP-complete)
L. Bulteau Pancake Flipping Is Hard 10/28
Reduction
L. Bulteau Pancake Flipping Is Hard 11/28
Efficient flips
A flip is efficient if it removes one breakpoint: π → π′
prd(πφ) = db(πφ) iff there exists a “path” of efficient flipsfrom πφ to the Identity. πφ →∗ I1
nAt most two efficient flips are possible from every permutation
x...h
h − 1...y
h + 1...
↙
h...x
h − 1...y
h + 1...
↘
y...
h − 1x...h
h + 1...
L. Bulteau Pancake Flipping Is Hard 12/28
Efficient flips
A flip is efficient if it removes one breakpoint: π → π′
prd(πφ) = db(πφ) iff there exists a “path” of efficient flipsfrom πφ to the Identity. πφ →∗ I1
nAt most two efficient flips are possible from every permutation
x...h
h − 1...y
h + 1...
↙
h...x
h − 1...y
h + 1...
↘
y...
h − 1x...h
h + 1...
L. Bulteau Pancake Flipping Is Hard 12/28
Efficient flips
A flip is efficient if it removes one breakpoint: π → π′
prd(πφ) = db(πφ) iff there exists a “path” of efficient flipsfrom πφ to the Identity. πφ →∗ I1
nAt most two efficient flips are possible from every permutation
x...h
h − 1...y
h + 1...
↙X
h...
x = h − 2h − 1...y
h + 1...
↘
y...
h − 1x...h
h + 1...
L. Bulteau Pancake Flipping Is Hard 12/28
Efficient flips
A flip is efficient if it removes one breakpoint: π → π′
prd(πφ) = db(πφ) iff there exists a “path” of efficient flipsfrom πφ to the Identity. πφ →∗ I1
nAt most two efficient flips are possible from every permutation
x...h
h − 1...y
h + 1...
↙
h...x
h − 1...
y = h + 2h + 1...
↘X
y...
h − 1x...h
h + 1...
L. Bulteau Pancake Flipping Is Hard 12/28
Efficient flips
A flip is efficient if it removes one breakpoint: π → π′
prd(πφ) = db(πφ) iff there exists a “path” of efficient flipsfrom πφ to the Identity. πφ →∗ I1
nAt most two efficient flips are possible from every permutation
x...h
h − 1...y
h + 1...
↙X
h...
x = h − 2h − 1...
y = h + 2h + 1...
↘X
y...
h − 1x...h
h + 1...
L. Bulteau Pancake Flipping Is Hard 12/28
Reduction ideas
Create πφ in order to know precisely which efficient flips arepossible (from πφ or subsequent permutations)
One possible flip: usual case, there is one path to followTwo possible flips: a choice has to be made
e.g., assigning “true” or “false” to a variableNo possible flip: bad choices have been made
e.g., a clause is unsatisfiedNecessity to end with the Identity permutation.
L. Bulteau Pancake Flipping Is Hard 13/28
Gadgets
Lock 3 states: closed, open, tested
Fork chooses between two optionsHook moves a subsequence in/out ofthe headDock stores subsequences when theyare sorted
Literals holds a lock for each literalin the formula
Variable opens locks correspondingto either x or ¬xClause tests one lock out of three
πφ
Literals Variable Clause
Lock Fork Hook Dock
Integers
L. Bulteau Pancake Flipping Is Hard 14/28
Gadgets
Lock 3 states: closed, open, testedFork chooses between two options
Hook moves a subsequence in/out ofthe headDock stores subsequences when theyare sorted
Literals holds a lock for each literalin the formula
Variable opens locks correspondingto either x or ¬xClause tests one lock out of three
πφ
Literals Variable Clause
Lock Fork Hook Dock
Integers
L. Bulteau Pancake Flipping Is Hard 14/28
Gadgets
Lock 3 states: closed, open, testedFork chooses between two optionsHook moves a subsequence in/out ofthe head
Dock stores subsequences when theyare sorted
Literals holds a lock for each literalin the formula
Variable opens locks correspondingto either x or ¬xClause tests one lock out of three
πφ
Literals Variable Clause
Lock Fork Hook Dock
Integers
L. Bulteau Pancake Flipping Is Hard 14/28
Gadgets
Lock 3 states: closed, open, testedFork chooses between two optionsHook moves a subsequence in/out ofthe headDock stores subsequences when theyare sorted
Literals holds a lock for each literalin the formula
Variable opens locks correspondingto either x or ¬xClause tests one lock out of three
πφ
Literals Variable Clause
Lock Fork Hook Dock
Integers
L. Bulteau Pancake Flipping Is Hard 14/28
Gadgets
Lock 3 states: closed, open, testedFork chooses between two optionsHook moves a subsequence in/out ofthe headDock stores subsequences when theyare sorted
Literals holds a lock for each literalin the formula
Variable opens locks correspondingto either x or ¬xClause tests one lock out of three
πφ
Literals Variable Clause
Lock Fork Hook Dock
Integers
L. Bulteau Pancake Flipping Is Hard 14/28
Gadgets
Lock 3 states: closed, open, testedFork chooses between two optionsHook moves a subsequence in/out ofthe headDock stores subsequences when theyare sorted
Literals holds a lock for each literalin the formula
Variable opens locks correspondingto either x or ¬x
Clause tests one lock out of three
πφ
Literals Variable Clause
Lock Fork Hook Dock
Integers
L. Bulteau Pancake Flipping Is Hard 14/28
Gadgets
Lock 3 states: closed, open, testedFork chooses between two optionsHook moves a subsequence in/out ofthe headDock stores subsequences when theyare sorted
Literals holds a lock for each literalin the formula
Variable opens locks correspondingto either x or ¬xClause tests one lock out of three
πφ
Literals Variable Clause
Lock Fork Hook Dock
Integers
L. Bulteau Pancake Flipping Is Hard 14/28
Lock gadget
Lock gadget (closed)
L =
129856431112
key = 10test = 7
Open
Lo =
12346589101112
Tested
I112 =
123456789101112
L. Bulteau Pancake Flipping Is Hard 15/28
Lock gadget
Lock gadget (closed)
L =
129856431112
key = 10test = 7
Open
Lo =
12346589101112
Tested
I112 =
123456789101112
L. Bulteau Pancake Flipping Is Hard 15/28
Lock gadget
Lock gadget (closed)
L =
129856431112
key = 10test = 7
Open
Lo =
12346589101112
Tested
I112 =
123456789101112
L. Bulteau Pancake Flipping Is Hard 15/28
Lock gadget
Lock gadget (closed)
L =
p + 1p + 2p + 9p + 8p + 5p + 6p + 4p + 3p + 11p + 12
key = p + 10test = p + 7
Open
Lo =
p + 1p + 2p + 3p + 4p + 6p + 5p + 8p + 9p + 10p + 11p + 12
Tested
Ip+1p+12 =
p + 1p + 2p + 3p + 4p + 5p + 6p + 7p + 8p + 9p + 10p + 11p + 12
L. Bulteau Pancake Flipping Is Hard 15/28
Lock gadget
Lock gadget (closed)
L =
129856431112
key = 10test = 7
Open
Lo =
12346589101112
Tested
I112 =
123456789101112
L. Bulteau Pancake Flipping Is Hard 15/28
Lock gadget
Opening
key...L...
→∗...Lo
...
Testing(when open)
test...Lo
...
→∗...I1
12...
Testing(when closed)
test...L...
→∗ ∅
L. Bulteau Pancake Flipping Is Hard 16/28
Lock gadget
Opening
key
XL
Y
→∗XLo
Y
Testing(when open)
test
XLo
Y
→∗XI1
12
Y
Testing(when closed)
test
XL
Y
→∗ ∅
L. Bulteau Pancake Flipping Is Hard 16/28
Lock gadget
keyXLY
∅↙
21
?X109856431112Y
↙
10X129856431112Y
↘
34658921
?X101112Y
↘
98564321
?X101112Y
↘
X12346589101112Y
=XLo
Y
L. Bulteau Pancake Flipping Is Hard 17/28
Lock gadget
keyXLY
∅↙
21
?X109856431112Y
↙
10X129856431112Y
↘
34658921
?X101112Y
↘
98564321
?X101112Y
↘
X12346589101112Y
=XLo
Y
L. Bulteau Pancake Flipping Is Hard 17/28
Lock gadget
keyXLY
∅↙
21
?X109856431112Y
↙
10X129856431112Y
↘
34658921
?X101112Y
↘
98564321
?X101112Y
↘
X12346589101112Y
=XLo
Y
L. Bulteau Pancake Flipping Is Hard 17/28
Lock gadget
keyXLY
∅↙
21
?X109856431112Y
↙
10X129856431112Y
↘
34658921
?X101112Y
↘
98564321
?X101112Y
↘
X12346589101112Y
=XLo
Y
L. Bulteau Pancake Flipping Is Hard 17/28
Lock gadget
keyXLY
∅↙
21
?X109856431112Y
↙
10X129856431112Y
↘
34658921
?X101112Y
↘
98564321
?X101112Y
↘
X12346589101112Y
=XLo
Y
L. Bulteau Pancake Flipping Is Hard 17/28
Lock gadget
testXLY
7X129856431112Y
→ ∅
L. Bulteau Pancake Flipping Is Hard 18/28
Lock gadget
testXLo
Y
∅↙
4321
?X76589101112Y
↙
7X12346589101112Y
↘
564321
?X789101112Y
↘
654321
?X789101112Y
↘
X123456789101112Y
=XI1
12Y
L. Bulteau Pancake Flipping Is Hard 19/28
Overall flow
Literalsxi : set Pi
¬xi : set Ni
Clause Cj
aj ∨ bj ∨ cj
I
Open locks in P1 Open locks in N1Open remaininglocks in P1 ∪ N1
......
......
Open locks in Pl Open locks in NlOpen remaininglocks in Pl ∪ Nl
Test lock a1 Test lock b1 Test lock c1Test remaining
locks in {a1, b1, c1}
......
......
Test lock ak Test lock bk Test lock ckTest remaining
locks in {ak , bk , ck}
L. Bulteau Pancake Flipping Is Hard 20/28
Fork gadget
E =
11873
F =
1096121345151421
F 1 =
109678111213141554321
(...)
F 2 =
378111096121345151421
(...)
Two efficient paths
XF 1
...
∗↙
EXF...
↘∗?XF 2
...
F 1
...→∗
?I115...
F 2
...→∗
?I115...
L. Bulteau Pancake Flipping Is Hard 21/28
Fork gadget
E =
11873
F =
1096121345151421
F 1 =
109678111213141554321
(...)
F 2 =
378111096121345151421
(...)
Two efficient paths
XF 1
...
∗↙
EXF...
↘∗?XF 2
...
F 1
...→∗
?I115...
F 2
...→∗
?I115...
L. Bulteau Pancake Flipping Is Hard 21/28
Fork gadget
E =
11873
F =
1096121345151421
F 1 =
109678111213141554321
(...) F 2 =
378111096121345151421
(...)
Two efficient paths
XF 1
...
∗↙
EXF...
↘∗?XF 2
...
F 1
...→∗
?I115...
F 2
...→∗
?I115...
L. Bulteau Pancake Flipping Is Hard 21/28
Fork gadget
E =
11873
F =
1096121345151421
F 1 =
109678111213141554321
(...) F 2 =
378111096121345151421
(...)
Two efficient paths
XF 1
...
∗↙
EXF...
↘∗?XF 2
...
F 1
...→∗
?I115...
F 2
...→∗
?I115...
L. Bulteau Pancake Flipping Is Hard 21/28
Hook gadget
G =34
H =
1211659821
take = 10
put = 7
G ′ =(...)
H ′ =(...)
G ′′ =(...)
H ′′ =(...)
Moves a substring up and down
take...GXH...
→∗
XG ′...H ′...
putX ′
G ′...H ′...
→∗
...G ′′
X ′
H ′′...
G ′′
XH ′′...
→∗X
?I112...
L. Bulteau Pancake Flipping Is Hard 22/28
Hook gadget
G =34
H =
1211659821
take = 10
put = 7
G ′ =(...)
H ′ =(...)
G ′′ =(...)
H ′′ =(...)
Moves a substring up and down
take...GXH...
→∗
XG ′...H ′...
putX ′
G ′...H ′...
→∗
...G ′′
X ′
H ′′...
G ′′
XH ′′...
→∗X
?I112...
L. Bulteau Pancake Flipping Is Hard 22/28
Hook gadget
G =34
H =
1211659821
take = 10
put = 7
G ′ =(...)
H ′ =(...)
G ′′ =(...)
H ′′ =(...)
Moves a substring up and down
take...GXH...
→∗
XG ′...H ′...
putX ′
G ′...H ′...
→∗
...G ′′
X ′
H ′′...
G ′′
XH ′′...
→∗X
?I112...
L. Bulteau Pancake Flipping Is Hard 22/28
Dock gadget
Dock(2, 7) =
1289
Stores a sorted substring (?I) out of thehead of the stack.
?I37...
Dock(2, 7)...
→∗...I1
9...
L. Bulteau Pancake Flipping Is Hard 23/28
Dock gadget
Dock(2, 7) =
1289
Stores a sorted substring (?I) out of thehead of the stack.
?I37...
Dock(2, 7)...
→∗...I1
9...
L. Bulteau Pancake Flipping Is Hard 23/28
Variable gadget
Variable gadget1 First part Move up the main sequence
2 Choose between xi and ¬xi
3 Open locks in Ni
4 Put back main sequence5 Other gadgets are activated6 Second part Hook collapses7 Open locks in Pi
8 Fork collapses9 Dock stores sorted sequences
10 End Gadget sorted
take
...
GE
keyp1
...keypq
putkeyn1
...keynq′
FH...
Dock
HookFork
⋆I⋆I...
Dock
L. Bulteau Pancake Flipping Is Hard 24/28
Variable gadget
Variable gadget1 First part Move up the main sequence2 Choose between xi and ¬xi
3 Open locks in Ni
4 Put back main sequence5 Other gadgets are activated6 Second part Hook collapses7 Open locks in Pi
8 Fork collapses9 Dock stores sorted sequences
10 End Gadget sorted
Ekeyp1
...keypq
putkeyn1
...keynq′
FG ′
...
H ′
...Dock
Hook
Fork
⋆I⋆I...
Dock
L. Bulteau Pancake Flipping Is Hard 24/28
Variable gadget
Variable gadget1 First part Move up the main sequence2 Choose between xi and ¬xi
3 Open locks in Ni
4 Put back main sequence5 Other gadgets are activated6 Second part Hook collapses7 Open locks in Pi
8 Fork collapses9 Dock stores sorted sequences
10 End Gadget sorted
keynq′...
keyn1
putkeypq
...keyp1
F 2
G ′
...
H ′
...Dock
Hook
⋆I⋆I...
Dock
L. Bulteau Pancake Flipping Is Hard 24/28
Variable gadget
Variable gadget1 First part Move up the main sequence2 Choose between xi and ¬xi
3 Open locks in Ni
4 Put back main sequence
5 Other gadgets are activated6 Second part Hook collapses7 Open locks in Pi
8 Fork collapses9 Dock stores sorted sequences
10 End Gadget sorted
putkeyn1
...keynq′
F 2
G ′
...
H ′
...Dock
Hook
⋆I⋆I...
Dock
L. Bulteau Pancake Flipping Is Hard 24/28
Variable gadget
Variable gadget1 First part Move up the main sequence2 Choose between xi and ¬xi
3 Open locks in Ni
4 Put back main sequence5 Other gadgets are activated
6 Second part Hook collapses7 Open locks in Pi
8 Fork collapses9 Dock stores sorted sequences
10 End Gadget sorted
...
G ′′
keyn1
...keynq′
F 2
H ′′
...Dock
Hook
⋆I⋆I...
Dock
L. Bulteau Pancake Flipping Is Hard 24/28
Variable gadget
Variable gadget1 First part Move up the main sequence2 Choose between xi and ¬xi
3 Open locks in Ni
4 Put back main sequence5 Other gadgets are activated6 Second part Hook collapses
7 Open locks in Pi
8 Fork collapses9 Dock stores sorted sequences
10 End Gadget sorted
G ′′
keyn1
...keynq′
F 2
H ′′
...Dock
Hook
⋆I⋆I...
Dock
L. Bulteau Pancake Flipping Is Hard 24/28
Variable gadget
Variable gadget1 First part Move up the main sequence2 Choose between xi and ¬xi
3 Open locks in Ni
4 Put back main sequence5 Other gadgets are activated6 Second part Hook collapses7 Open locks in Pi
8 Fork collapses9 Dock stores sorted sequences
10 End Gadget sorted
keyn1
...keynq′
F 2
⋆I...
Dock
⋆I⋆I...
Dock
L. Bulteau Pancake Flipping Is Hard 24/28
Variable gadget
Variable gadget1 First part Move up the main sequence2 Choose between xi and ¬xi
3 Open locks in Ni
4 Put back main sequence5 Other gadgets are activated6 Second part Hook collapses7 Open locks in Pi
8 Fork collapses
9 Dock stores sorted sequences10 End Gadget sorted
F 2
⋆I...
Dock
⋆I⋆I...
Dock
L. Bulteau Pancake Flipping Is Hard 24/28
Variable gadget
Variable gadget1 First part Move up the main sequence2 Choose between xi and ¬xi
3 Open locks in Ni
4 Put back main sequence5 Other gadgets are activated6 Second part Hook collapses7 Open locks in Pi
8 Fork collapses9 Dock stores sorted sequences
10 End Gadget sorted
⋆I⋆I...
Dock
L. Bulteau Pancake Flipping Is Hard 24/28
Variable gadget
Variable gadget1 First part Move up the main sequence2 Choose between xi and ¬xi
3 Open locks in Ni
4 Put back main sequence5 Other gadgets are activated6 Second part Hook collapses7 Open locks in Pi
8 Fork collapses9 Dock stores sorted sequences
10 End Gadget sorted
...I
⋆I⋆I...
Dock
L. Bulteau Pancake Flipping Is Hard 24/28
Clause gadget
Same structure,with twochoices:[[a or b] or c]
take1
...G1
E1
take2
put1testcF1
G2
E2
testaput2testbF2
H2
H1
...Dock1
Dock2
Hook 1
Hook 2
Fork 1
Fork 2
L. Bulteau Pancake Flipping Is Hard 25/28
Overall construction
πφ =
takeV1...
takeVl
takeC1...
takeCk
V1...VlC1...Ck
(docks)(locks)
I
Open locks in P1 Open locks in N1Open remaininglocks in P1 ∪ N1
......
......
Open locks in Pl Open locks in NlOpen remaininglocks in Pl ∪ Nl
Test lock a1 Test lock b1 Test lock c1Test remaining
locks in {a1, b1, c1}
......
......
Test lock ak Test lock bk Test lock ckTest remaining
locks in {ak , bk , ck}
L. Bulteau Pancake Flipping Is Hard 26/28
Finally
There exists an efficient path from πφ to the identity iff φ issatisfiable.The construction requires a polynomial time.MIN-SBPR is NP-hard.
L. Bulteau Pancake Flipping Is Hard 27/28
Conclusion
The complexity class of the Pancake Flipping problem is settledThere remains many intriguing questions:
What about the burnt variant?Any approximation algorithm?Any FPT algorithm with a relevant parameter?Any better bound for the diameter than1.07n ≤ f (n) ≤ 1.64n (unburnt) and1.5n ≤ g(n) ≤ 2n (burnt)?
Thank you!
L. Bulteau Pancake Flipping Is Hard 28/28
Conclusion
The complexity class of the Pancake Flipping problem is settledThere remains many intriguing questions:
What about the burnt variant?Any approximation algorithm?Any FPT algorithm with a relevant parameter?Any better bound for the diameter than1.07n ≤ f (n) ≤ 1.64n (unburnt) and1.5n ≤ g(n) ≤ 2n (burnt)?
Thank you!
L. Bulteau Pancake Flipping Is Hard 28/28