Combinatorial Search

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Combinatorial Search

• Spring 2008 (Fourth Quarter)

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Some practical remarks

• 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 year, 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 Kompendium available at GAD online
  notes.

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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 group
  dSoegOpt Or try finding someone after the
  class!
• Exercise classes are on specific dates, see
  webpage

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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 cant 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 By Branching

TSP

Efficient 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
Mixed Integer Linear Programming
TSP

Exponential
No reduction, unless PNP
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, non-members 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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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 60161300000101200
04001400090020
000704000000
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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 ltx, b(j), ygt 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 lt desc(F), desc(f), b(?), b(?) gt
• 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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Some Turing Machine Applets
http//math.hws.edu/TMCM/java/labs/xTuringMachineL
ab.html
http//www.igs.net/tril/tm/tm.html
http//web.bvu.edu/faculty/schweller/Turing/Turing
.html
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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 60161300000101200
04001400090020
000704000000
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java MaxFlow 111111 111111111111111111111111111
11 1111111111111111111111 1111111111
11111111 11111111111111111111111111111
11111111111
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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 PNP? 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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Solved (Perelman)!
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P vs. NP and mathematics

• If PNP, mathematicians may be replaced by (much
  more reliable) computers
• PNP ) 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 PNP.

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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 PNP.