Transcript Traveling Salesman Problem

Combinatorial Search
Spring 2010 (Q4)
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Some practical remarks
• Lecturers: Kristoffer Arnsfelt Hansen and Peter Bro Miltersen
• Homepage: www.daimi.au.dk/dKS
• Exam: Oral, 7-scale 30 minutes, including overhead, no preparation.
• There will be three compulsory assignments. If you want to transfer
credit from last years, let me know as soon as possible and before
May 1.
• The solution to the compulsory assignments should be handed in at
specific exercise sessions and given to the instructor in person.
• Text: Virtual handouts
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Exercise classes
• You can switch classes iff you find someone to switch
with.
• To find someone, post an add in the forum on the
webpage. Or try finding someone after the class!
• Exercise classes start next Thursday.
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Frequently asked questions about
compulsory assignments
•
•
Q: Do I really have to hand in all three assignments?
A: YES!
•
•
Q: Do I really have to hand in all three assignments on time?
A: YES!
•
•
Q: What if I can’t figure out how to solve them?
A: Ask your instructor. Start solving them early, so that you will have
sufficient time.
•
•
Q: What if I get sick or my girlfriend breaks up or my hamster dies?
A: Start solving them early, so that you will have sufficient time in case of
emergencies.
•
•
Q: Do I really have to hand in all three assignments?
A: YES!
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“Optimization” – a summary!
Mixed Integer Linear Programming
Exponential algorithm
by Branching
TSP
Efficient algorithm
by Local Search
…
…
?
Linear Programming
Min Cost Flow
= reduction
Max Flow
Maximum matching
Shortest paths
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Lots and lots of man hours….
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NP-completeness
Mixed Integer Linear Programming
Exponential
By Branching
…
TSP
Efficient
by Local Search
…
?
Linear Programming
Min Cost Flow
= reduction
Max Flow
Maximum matching
Shortest paths
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NP-completeness
…
Exponential
Mixed Integer Linear Programming
TSP
No reduction, unless P=NP
Efficient
by Local Search
Linear Programming
Min Cost Flow
= reduction
Max Flow
Maximum matching
Shortest paths
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Very useful for saving (human) time
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Rigorous Formalization
“Problems”
Languages
“Efficient Algorithms”
Turing Machines, P
“Search Problems”
NP
“Reductions”
Polynomial Reductions
“Universal Search Problems”
NPC
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Problems: Languages
• A language L is a subset of {0,1}*.
• A language models a decision problem:
Members of L are the yes-instances, nonmembers are the no-instances.
• This is the only kind of problem our theory shall
be concerned with!
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Restriction: Inputs Boolean Strings
• Strings over an arbitrary alphabet can be
represented as Boolean Strings (Ex: ascii,
unicode).
• In reality, computers may only hold Boolean
strings (their memory image is a bit string).
• Arbitrary real numbers may not be represented
but this is intentional!
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How to encode max flow instance?
java MaxFlow 6#0|16|13|0|0|0#0|0|10|12|0|0
#0|4|0|0|14|0#0|0|9|0|0|20
#0|0|0|7|0|4|#0|0|0|0|0|0
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Models of Computation
• Model 1: Our computer holds exact real numbers. The size of the
input is the number of real numbers in the input. The time complexity
of an algorithm is the number of arithmetic operations performed.
• Model 2: Our computer holds bits and bytes. The size of the input is
the number of bits in the input. The time complexity of an algorithm
is the number of bit-operations performed.
• We know an efficient algorithm for linear programming in Model 2
but not model 1.
• The NP-completeness theory is intended to capture Model 2
and not Model 1.
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How to encode max flow instance?
java MaxFlow 6#0|16|13|0|0|0#0|0|10|12|0|0
#0|4|0|0|14|0#0|0|9|0|0|20
#0|0|0|7|0|4|#0|0|0|0|0|0
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Restriction: All inputs “legal”.
• Any string should be either a yes-string or
a no-string.
• It would be nice to also have “malformed”
strings.
• However, we shall just lump the
malformed strings with the no-strings.
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Restriction: Output yes or no
• Suppose we want to consider computing a
function, f: {0,1}* ! {0,1}*.
• Stand-in for f:
Lf = {<x, b(j), y> | f(x)j = y}
• Lf has an efficient algorithm if and only if f
has an efficient algorithm.
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Restriction: functions
• OPT: Given description of F, f find x 2 F
maximizing f(x).
• There may be several optimal solutions.
OPT does not seem to be captured easily by a
function.
• Stand-in for OPT:
LOPT = { < desc(F), desc(f), b(), b() >
| some x 2 F has f(x) ¸  /  }
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LOPT vs. OPT
• LOPT may be easy to solve even though
OPT is hard to solve, so LOPT is not a
perfect stand-in.
• However, if LOPT has no efficient solution,
then OPT has no efficient solution, so LOPT
can still be used to argue that OPT is hard.
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Algorithms: Turing Machines
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Turing Machines
• A Turing machine consists of an infinite tape,
divided into cells, each holding a symbol from
alphabet  that includes 0,1,#.
• A tape head is at any point in time positioned at
a cell.
• A finite control reads the symbol at the head,
updates the symbol at the head and the position
of the head.
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Finite Control
• Finite set of states Q. The control is in exactly
one of the state. Three special states: start,
accept, reject.
• Transition function:
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Running the machine on an input
• The input string is placed on the tape (surrounded by
blanks) and the head positioned to the immediate left of
the input. The initial state of the finite control is start.
• If the finite control eventually goes to accept state, the
input is accepted (“the machine outputs yes”).
• If the finite control eventually goes to reject state, the
input is rejected (“the machine outputs no”).
• The machine is said to decide a language L if it accepts
all members of L and rejects all members of {0,1}*-L.
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A nice Turing Machine Applets
http://ironphoenix.org/tril/tm/
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If you want to make
rigorous the notion of an
“efficient algorithm” why
do you choose such a
hopelessly inefficient
device ??!?
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Church-Turing Thesis
Any decision problem that can be solved
by some mechanical procedure, can be
solved by a Turing machine.
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Polynomial Church-Turing thesis
A decision problem can be solved in
polynomial time by using a reasonable
sequential model of computation if and
only if it can be solved in polynomial time
by a Turing machine.
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The complexity class P
• P := the class of decision problems (languages) decided
by a Turing machine so that for some polynomial p and
all x, the machine terminates after at most p(|x|) steps on
input x.
• By the Polynomial Church-Turing Thesis, P is “robust”
with respect to changes of the machine model.
• Is P also robust with respect to changes of the
representation of decision problems as languages?
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How to encode max flow instance?
java MaxFlow 6#0|16|13|0|0|0#0|0|10|12|0|0
#0|4|0|0|14|0#0|0|9|0|0|20
#0|0|0|7|0|4|#0|0|0|0|0|0
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java MaxFlow 111111
#|1111111111111111|1111111111111|||
#||1111111111|111111111111||
#|1111|||11111111111111|
#||111111111|||11111111111111111111
#|||1111111||1111 #|||||
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Polynomial time computable maps
f: {0,1}* ! {0,1}* is called polynomial time
computable if for some polynomial p,
- For all x, |f(x)| · p(|x|).
- Lf 2 P.
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Polynomially equivalent
representations
• A representation of objects (say graphs, numbers) as
strings is good if the language of valid representations is
in P.
• Two different representations of objects are called
polynomially equivalent if we may translate between
them using polynomial time computable maps.
• Ex: Adjacency matrices vs. Edge lists
• Ex: Binary vs. Decimal
• Counterexample: Binary vs. Unary
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Robustness of Representation
• Given two good, polynomially equivalent
representations of the instances of a
decision problem, resulting in languages
L1 and L2 we have
L1 2 P iff L2 2 P.
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Rigorous Formalization
“Problems”
Languages
“Efficient Algorithms”
Turing Machines, P
“Search Problems”
NP
“Reductions”
Polynomial Reductions
“Universal Search Problems”
NPC
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Search Problems: NP
• We want to capture decision problems that can
be solved by exhaustive search of the following
kind.
• Go through a space of possible solutions,
checking for each one of them if it is an actual
solutions (in which case the answer is yes).
• Example: Compositeness. Given a number in
binary, is it a product of smaller numbers?
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Search Problems: NP
L is in NP iff there is a language L’ in P and a
polynomial p so that:
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Intuition
• The y-strings are the possible solutions
to the instance x.
• We require that solutions are not too long
and that it can be checked efficiently if a
given y is indeed a solution (we have a
“simple” search problem)
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P vs. NP
• P is a subset of NP
• Is P=NP? Then any “simple” search
problem has a polynomial time algorithm.
• This is the most famous open problem
of mathematical computer science!
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.Clay Mathematics
Institute
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Announced 16:00, on Wednesday, May 24, 2000
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P vs. NP and mathematics
• If P=NP, mathematicians may be replaced by
(much more reliable) computers:
P=NP ) There is an algorithmic procedure that
takes as input any formal math statement and
always outputs its shortest formal proof in time
polynomial in the length of the proof.
• This is usually regarded (in particular by
mathematicians!) as evidence that P and NP are
different.
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Rigorous Formalization
“Problems”
Languages
“Efficient Algorithms”
Turing Machines, P
“Search Problems”
NP
“Reductions”
Polynomial Reductions
“Universal Search Problems”
NPC
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Reductions
• A reduction r of L1 to L2 is a polynomial time
computable map so that
8 x: x 2 L1 iff r(x) 2 L2
• We write
L1 · L2
if L1 reduces to L2.
• Intuition: Efficient software for L2 can also be
used to efficiently solve L1.
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Properties of reductions
Transitivity:
L1 · L2 Æ L2 · L3 ) L1 · L3
Downward closure of P:
L1 · L2 Æ L2 2 P ) L1 2 P.
• Follows from Polynomial Church-Turing
thesis.
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NP-hardness
• A language L is called NP-hard iff
8 L’ 2 NP: L’ · L
• Intuition: Software for L is strong enough
to be used to solve any simple search
problem.
• Proposition: If some NP-hard language is
in P, then P=NP.
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NPC
• A language L 2 NP that is NP-hard is
called NP-complete.
• NPC := the class of NP-complete
problems.
• Proposition:
L 2 NPC ) [L 2 P iff P=NP].
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