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Title: Explorations in Artificial Intelligence


1
Explorations in Artificial Intelligence
  • Prof. Carla P. Gomes
  • gomes_at_cs.cornell.edu
  • Module 6
  • Intro to Complexity

2
The algorithm problem
Any legal input
Specification of all legal inputs
and
The algorithm
Specification of desired output as a function of
the input
The desired output
3
Examples of algorithmic problems
4

Examples of algorithmic problems
5
Variants of algorithmic problems Decision Problem
Problem Knapsack (decision) Input profits p0,
p1, , pn-1 weights w0, w1, ,
wn-1 capacity M target profit P Output YES
or NO to the question Does there exist an
n-tuple xo,,x n-1 ? 0,1n, such that

and
6
Variants of algorithmic problems Search Problem
Problem Knapsack (search) Input profits p0,
p1, , pn-1 weights w0, w1, , wn-1 capacity
M target profit P Output An n-tuple xo,,x
n-1 ? 0,1n, such that

and
7
Variants of algorithmic problems Optimal Value
Problem Knapsack (optimal value) Input profits
p0, p1, , pn-1 weights w0, w1, ,
wn-1 capacity M Output The maximum value of

subject to
and xo,,x n-1 ? 0,1n
8
Variants of algorithmic problems Optimization
Problem Knapsack (optimization) Input profits
p0, p1, , pn-1 weights w0, w1, ,
wn-1 capacity M Output An n-tuple xo,,x
n-1 ? 0,1n, such that

is maximized subject to
9
Instance of an algorithmic problemSize of an
instance
  • An instance of an algorithmic problem is a
    concrete case of such a problem with specific
    input. The size of an instance is given by the
    size of its input.
  • Examples of instances
  • An instance of problem 1

Size of instance ? length of list
Size of instance L 7
10
Examples of instances
Size of instance ? Number of cities and roads

A particular instance
Size of instance 6 nodes 9 edges
The size of an instance is given by the size of
its input.
11

The size of an instance is given by the size of
its input.
Size of instance ? Number of variables and
constraints (n,m)
An instance of problem 5 max 3 x 5 y s.t
x 4 y 12 3 x 2 y 18 x,y 0
Size of instance 2 variables 3 functional
constraints
12
Complexity of Algorithms
13
Complexity of Algorithms
  • The complexity of an algorithm is the number of
    steps that it takes to transform the input data
    into the desired output.
  • Each simple operation (,-,,/,,if,etc) and each
    memory access corresponds to a step.()
  • The complexity of an algorithm is a function of
    the size of the input (or size of the instance).
    Well denote the complexity of algorithm A by
    CA(n), where n is the size of the input.

() This model is a simplification but still
valid to give us a good idea of the complexity of
algorithms.
14
Example Insertion Sort
From Introduction to Algorithms Cormen et al
15
Different notions of complexity
Worst case complexity of an algorithm A the
maximum number of computational steps required
for the execution of Algorithm A, over all the
inputs of the same size, s. It provides an upper
bound for an algorithm. The worst that can happen
given the most difficult instance the
pessimistic view.

Best case complexity of an algorithm A -the
minimum number of computational steps required
for the execution of Algorithm A, over all the
inputs of the same size, s. The most optimistic
view of an algorithm it tells us the least work
a particular algorithm could possibly get away
with for some one input of a fixed size we have
the chance to pick the easiest input of a given
size.  
16
  • Average case complexity of an algorithm A - i.e.,
    the average amount of resources the algorithm
    consumes assuming some plausible frequency of
    occurrence of each input.
  • Figuring out the average cost is much more
    difficult than figuring out either the worst-cost
    or best-cost ? e.g., we have to assume a given
    probability distribution for the types of inputs
    we get.

 
17
Different notions of complexity
We perform upper bound analysis on algorithms.
18
Growth Rates
  • In general we only worry about growth rates
    because
  • Our main objective is to analyze the cost
    performance of algorithms.
  • Another obstacle to having the exact cost of
    algorithms is that sometimes the algorithm are
    quite complicated to analyze.
  • When analyzing an algorithm we are not that
    interested in the exact time the algorithm takes
    to run often we only want to compare two
    algorithms for the same problem the thing that
    makes one algorithm more desirable than another
    is its growth rate relative to the other
    algorithms growth rate.

19
Growth Rates
  • Two functions of n have different growth rates if
    as n goes to infinity their ratio either goes to
    infinity or goes to zero.
  • If their ratio stays near a non-zero constant
    then they are asymptotically the same function.

20
Big Oh Notation
  • Given two functions F and G, whose domain is the
    natural numbers, we say that the order of F is
    lower than or equal to the order of G if
  • F(n) c G(n) for all n gt n0 (c and n0 are
    constants)
  • We say F is O(G) (F is oh of G)

Example (3 n3 n2 n ) is O(n3)
In practice we just look at the fastest growing
term of the expression
21
Typical Growth Rates
22
Roughly Speaking
exponential
quadratic
Cost
linear
logarithmic
constant
Size
23
Good vs. Bad Algorithms
24
  • How do computer scientists differentiate between
  • good (efficient) and bad (not efficient)
    algorithms?

The yardstick is that any algorithm that runs in
no more than polynomial time is an efficient
algorithm everything else is not.
25
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26

Polynomial vs. exponential growth (Harel 2000)
N2
27
Problem Complexity
  • Theory of NP-completeness or NP-hardness
  • Easy vs. hard problems

28
Overview of complexity
  • How can we show a problem is efficiently
    solvable?
  • We can show it constructively. We provide an
    algorithm and show that it solves the problem
    efficiently. E.g.
  • Shortest path problem - Dijkstras algorithm runs
    in polynomial time, O(n2). Therefore the shortest
    path problem can be solved efficiently.
  • Linear Programming The Interior Point method
    has polynomial worst-case complexity. Therefore
    Linear programming can be solved efficiently.

() The simplex method has exponential worst case
complexity/ However, in practice the simplex
algorithm seems to scale as m3, where m is the
number of functional constraints.
29
Overview of complexity
  • How can we show a problem is not efficiently
    solvable?
  • How do you prove a negative? Much harder!!!
  • This is the aim of complexity theory.

30
Easy (efficiently solvable) problems vsHard
Problems
  • Easy Problems - we consider a problem X to be
    easy or efficiently solvable, if there is a
    polynomial time algorithm A for solving X. We
    denote by P the class of problems solvable in
    polynomial time.
  • Hard problems --- everything else. Any problem
    for which there is no polynomial time algorithm
    is an intractable problem.
  • .

31
Class P Class of Problems solvable in Polynomial
Time
Other examples of problems in the class P
include The assignment problem and
transportation problem, finding the minimum cost
flow problem and the max cost flow in in a
directed graph.
32
Two problems
33
Id like you to develop an effcient algorithm to
find the longest path between two points in a
graph.
34
Your Longest Path Algorithm between two nodes, u
and v
G(N,E)

u
v
Initialization MaxPath ? none MaxPathLength
? 0 For each path P starting at 1 if P is a
simple path from u to v and length(P) gt
MaxPath MaxPath ? P MaxPathLength ?
length(P) Return MaxPath MaxPathLength
35
Is that the best you can do? -- that seems to be
a bad algorithm!!!
36
I cant find an efficient algorithm. I guess Im
too dumb.
37
I cant find an efficient algorithm, but neither
can these famous researchers.
38
In 1936, Alan Turing, a British mathematician,
showed that there exists a relatively simple
universal computing device that can perform any
computational process. Computers use such a
universal model.
Turing Machine (abstraction)
Turing also showed the limits of computation
some problems cannot be computed even with the
most powerful computer and even with unlimited
amount of time e.g., Halting problem.
39
Brilliant mathematician, synthesizer, and
promoter of the stored program concept, whose
logical design of the Institute of Advanced
Studies (Princeton) Computer became the prototype
of todays computer () - the von Neumann
Architecture.
() sequential i.e., non-parallel computers
40
Invented the theory of NP-Completeness proved
that a simple problem - Satisfiability is
NP-Complete.
Given a propositional formula, is there an
assignment to its variables (a, b, and c True
or False) making the formula true?
Showed that several important problems and
applications are NP-Complete and
NP-hard, including Integer Programming.
41
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42
Invented Linear Programming Formulations max
3 x 5 y s.t x 4 y 12 3 x 2 y
18 x,y 0 Invented Simplex Algorithm
43
Theory of NP-completeness and NP-hardnessEasy
vs. hard problems
44
Can satisfiability or integer programming be
solved in polynomial time?
  • FACT every algorithm that has ever been
    developed for satisfiability or integer
    programming takes exponential time.
  • Hundreds of very smart researchers have tried to
    come up with polynomial time algorithms for
    satisfiability or integer programming, and
    failed.
  • It is generally believed that there is no
    polynomial time algorithm for satisifiability or
    integer programming.
  • Complexity theory deals with proving that
    satisfiability, integer programming, and many
    other problems are computationally hard.

45
Decision Problems
NP-Completeness theory deals with decision
problems. What is a decision problem? A problem
for which there is a yes/no answer. Examples
?Is there a path between two nodes in a graph
shorter than k? (decision version of shortest
path problem) ?Is there a path between two nodes
in a graph longer than k? (decision version of
longest path problem) ?Most optimization
problems can be formulated as a decision problem
46
Class NP Class of Problems solvable in
Nondeterministic-Polynomial Time
  • We say that a decision problem is solvable in
    Non-deterministic polynomial time if
  • The solution can be verified in polynomial time.
    (E.g., verifying that a path has length greater
    than K)
  • If we imagine that we have an exponential number
    of processors, we can check all possible
    solutions simultaneously and therefore answer in
    polynomial time

47
Class NP-Complete
The first problem to be shown to be NP-Complete
was Satisfiability Cook showed that all the
problems in NP could be translated (in polynomial
time) as Satisfiability problems The word
complete means that every problem in the class
NP-complete can be reduced (in polynomial time)
into another problem of the class NP-complete.
For example all the problems in the class
NP-complete can be written as Satisfiability
problems. The class of NP-Complete problems is
the class of the hardest computational problems
in the class NP every NP problem can be
transformed into an NP-complete problem (the
reverse is not true!!!)
48
Is P not equal to NP?1,000,000 question
  • P not equal to NP?
  • Is that true that not all problems is NP can be
    solved in polynomial time?
  • Class of NP-Complete Problems
  • They all admit exponential time solutions
  • Nobody has ever been able to find a polynomial
    time solution for any single problem in the
    class
  • Nobody has ever been able to prove an exponential
    lower bound for any single problem in the class

49
P not equal to NP?1,000,000 question
  • Pictorial interpretation of this question

Is this the right picture? There are
problems in NP that are inherently
intractable and cannot be solved in polynomial
time.
50
P not equal to NP?1,000,000 question
  • Pictorial interpretation of this question

Or is this the right picture? All the
problems in NP can be solved in polynomial time.
Even though at this point we dont know of
polynomial time algorithms to solve some
problems in NP, they exist
P NP
51
P vs. NPOne Million Dollar Prize

http//www.claymath.org/Millennium_Prize_Problems/
P_vs_NP/
52
Class of NP-Complete ProblemsOne Million Dollar
Prize
  • Completeness
  • if someone were to find a polynomial time
    solution for a single problem in the class
    NP-complete ? all the problems could be solved in
    polynomial time!!!
  • if someone were to prove an exponential lower
    bound for a single problem in the class
    NP-complete ? all the problems in the class would
    be intractable !!!

53
P vs. NPOne Million Dollar Prize

http//www.claymath.org/Millennium_Prize_Problems/
P_vs_NP/
54
Class of NP-Complete ProblemsOne Million Dollar
Prize
  • Completeness
  • if someone were to find a polynomial time
    solution for a single problem in the class
    NP-complete ? all the problems could be solved in
    polynomial time!!!
  • if someone were to prove an exponential lower
    bound for a single problem in the class
    NP-complete ? all the problems in the class would
    be intractable !!!

55
  • Conjecture
  • NP-Complete problems are inherently hard!
  • They are intractable ?!

56
On Proving NP-Completeness results
  • Suppose that we want to prove that the a problem
    ?
  • is NP-Complete. How do we do it?
  • Find a known NP-complete problem, ?NPC.
  • Show that ?NPC can be transformed (in polynomial
    time) into ?.
  • Show that there is a solution to ?NPC if and
    only if there there is a solution to ?.

57
Proof Hamiltonian Path is NP-complete
  • A hamiltonian cycle is a cycle that passes
    through each node exactly once.
  • A hamiltonian path is a path that includes every
    node of G.
  • Suppose that we know that the problem of deciding
    if there is a hamiltonian cycle in a graph is
    NP-Complete.
  • We will show that the problem of deciding if
    there is a hamiltonian path is also NP-Complete.

58
Proof Technique
  • Start with any instance of the hamiltonian cycle
    problem (?NPC).
  • We denote this instance as G (N, A).
  • Transformation proofs (these are standard).
    Create an instance G (N, A) for the
    hamiltonian path problem from G with the
    following property there is a hamiltonian path
    in G if and only if there is a hamiltonian cycle
    in G.

59
The transformed network node 1 of the original
network was split into nodes 1 and 21, and nodes
0 and 22 were connected to the split nodes.
The original network
From J.Orlin
60
Claim If there is a hamiltonian cycle in the
original graph then there is a hamiltonian path
in the transformed graph.
22
1
21
0
1
A Hamiltonian Cycle.
Take the two arcs in G incident to the node 1.
Connect one to node 1, and the other to node 21.
Add in arcs (0,1) and (21, 22).
61
Claim If there is a hamiltonian path in the
transformed graph then there is a hamiltonian
cycle in the original graph.
22
1
21
0
1
A Hamiltonian Path
Delete the two arcs (0, 1) and (21, 22). Then
take the other arcs in G incident to 1 and 21,
and make them incident to node 1 in G.
62
NP-CompletenessLongest Path
Problem Longest Path Input Graph G (V,E),
length l for each edge, positive integer K
V. Output Is there a simple path (i.e., a path
visiting each node at most once) with K or more
edges?
Problem HamiltonianPath Input Graph G
(V,E). Output Does it have a Hamiltonian path
(i.e., a path visiting each node exactly once)?
63
Reduction NP-CompletenessFrom Hamiltonian Path
into Longest Path
HamiltonianPath is NP-Complete
Problem HamiltonianPath Input Graph G
(V,E). Output Does it have a Hamiltonian path
(i.e., a path visiting each node exactly once)?
Problem Longest Path Input Graph G (V,E),
length l for each edge, positive integer K
V. Output Is there a simple path (i.e., a path
visiting each node at most once) with K or more
edges?
If there is a path of length K V (Yes)
64
Reduction NP-CompletenessFrom Hamiltonian Path
into Longest Path
Problem HamiltonianPath Input Graph G
(V,E). Output Does it have a Hamiltonian path
(i.e., a path visiting each node exactly once)?
Problem Longest Path Input Graph G (V,E),
length l for each edge, positive integer K
V. Output Is there a simple path (i.e., a path
visiting each node at most once) with K or more
edges?
If there is not a path of length K V (No)
65

Reduction NP-CompletenessFrom Hamiltonian Path
into Longest Path
Problem HamiltonianPath Input Graph G
(V,E). Output Does it have a Hamiltonian path
(i.e., a path visiting each node exactly once)?
If there is a Hamiltonian Path in G (YES).
Problem Longest Path Input Graph G (V,E),
length l for each edge, positive integer K
V. Output Is there a simple path (i.e., a path
visiting each node at most once) with K or more
edges?
66
Reduction NP-CompletenessFrom Hamiltonian Path
into Longest Path
Problem HamiltonianPath Input Graph G
(V,E). Output Does it have a Hamiltonian path
(i.e., a path visiting each node exactly once)?
If there is NOT a Hamiltonian Path in G (NO).
Problem Longest Path Input Graph G (V,E),
length l for each edge, positive integer K
V. Output Is there a simple path (i.e., a path
visiting each node at most once) with K or more
edges?
67
Proof of NP-Completeness of Sudoku
  • Suppose that we know that the problem of deciding
    if we can complete a Latin Square is
    NP-Complete i.e., the Latin Square Completion
    problem is NP-Complete.
  • We will show that the problem of deciding if we
    can complete a partial Sudoku instance is also
    NP-Complete.

68
Sudoku
Can we complete this matrix using numbers from 1
to 9, with repeating a number in a row, column,
or block?
9 55 3x 10 52 possible completions
69
0
1
0
1
3
5
6
8
4
7
Reduction To Sudoku
1
3
5
6
8
4
7
0
1
6
8
7
0
1
3
5
4
1
2
7
8
6
4
3
5
1
Latin Square Completion Problem
4
5
3
1
7
8
6
4
5
3
1
7
8
6
1
2
5
3
4
8
6
7
1
2
5
3
4
8
6
7
1
2
8
6
7
5
3
4
70
If there is a Latin Square Completion There
is also a way of Completing the Sudoku matrix
0
1
0
1
3
5
6
8
4
7
2
0
1
1
3
5
6
8
4
7
0
1
2
1
6
8
7
0
1
3
5
4
2
1
2
1
2
7
8
6
4
3
5
1
2
0
Latin Square Completion Problem
0
2
4
5
3
1
7
8
6
4
5
3
1
2
0
7
8
6
1
2
5
3
4
8
6
7
0
1
2
5
3
4
8
6
7
0
1
2
5
3
4
8
6
7
0
71
There is also a way of completing the Latin
Square Completion If there is a way of
completing the Sudoku matrix
0
1
0
1
3
5
6
8
4
7
2
1
3
5
6
8
4
7
0
1
2
6
8
7
0
1
3
5
4
2
1
2
7
8
6
4
3
5
1
2
0
Latin Square Completion Problem
0
2
4
5
3
1
7
8
6
4
5
3
1
2
0
7
8
6
1
2
5
3
4
8
6
7
0
1
2
5
3
4
8
6
7
0
1
2
5
3
4
8
6
7
0
72
Class NP vs. Class Co-NP
Class NP ? Class of decision problems whose
solution can be verified in polynomial time.
(E.g., satisfiability) Class Co-NP ? co-NP is the
complexity class that contains the complements of
decision problems in the complexity class NP.
73
Co-NP and the asymmetry of NP
Unsatisfiability (Complement of
Satisfiability)- Is it true that for all values
of a, b, and c this formula is not
satisfiable? YES instances dont have short
answers If indeed the answer to this question
is YES (i.e., we have an unsatisfiable clause)
we do not have a short proof for it.
Class Co-NP ? co-NP is the complexity class that
contains the complements of decision problems
in the complexity class NP.
74
Class P vs. NP ? Co-NP
If a problem belongs to P, then it belongs to
both NP and co-NP, so P ? NP ? Co-NP
P NP ? Co-NP ? (i.e., are there problems with
good characterization but with no polynomial time
algorithm?)
75
Class PSPACE
  • What if we worry about space - i.e., memory
  • requirements?
  • Class PSPACE - the set of decision problems
  • that can be solved using a polynomial amount
  • of memory, and unlimited time.
  • Clearly P ? PSPACE

76
Class PSPACE
  • Class PSPACE - class of problems that appears
  • to be harder than NP and co-NP.
  • Why? space can be re-used while time cannot.
  • Examples
  • Consider an algorithm that counts from 0 to 2n
    1. while this algorithm can be implemented with a
    simple n-bit counter, it runs for an exponential
    time!
  • We can also solve the satisfiability problem
    using only a polynomial amount of space, for
    example by trying all possible assignments using
    also an n-bit counter.

77
PSPACE-Complete
  • A decision problem is in PSPACE-complete if it
    is in PSAPCE, and
  • every problem in PSPACE can be reduced to it in
    polynomial time. The
  • problems in PSPACE-complete can be thought of as
    the hardest problems
  • in PSPACE. These problems are widely suspected to
    be outside of P and
  • NP, but that is not known.

78
Satisfiability (NP-complete)
Quantified Satisfiability (PSPACE-complete) (The
most basic PSPACE-complete problem is identical
to satisfiability, except it alternates
existential and universal quantifiers)

The PSPACE-complete problem resembles a game is
there some move I can make, such that for all
moves my opponent might make, there will then be
some move I can make to win? The question
alternates existential and universal
quantifiers. Not surprisingly, many puzzles turn
out to be NP-complete, and many games turn out
to be PSPACE-complete.
79
  • Checkers is PSPACE-complete when generalized so
    that it can be played on an
  • n n board.
  • Generalized versions of the games Hex and Reversi
    and
  • the solitaire games such as Rush Hour, Mahjong,
    Atomix and Sokoban.
  • Note that the definition of PSPACE-complete is
    based on
  • asymptotic complexity the time it takes to solve
    a
  • problem of size n, in the limit as n grows
    without bound.
  • That means a game like checkers (which is played
    on an 8
  • 8 board) could never be PSPACE-complete. That
    is why
  • all the games were modified by playing them on an
    n n
  • board instead.

80
PSPACE
Co-NP
NP
P
81
Some examples of NP-hard problems
  • Longest path
  • Traveling Salesman Problem
  • Capital Budgeting Problem (knapsack problem)
  • Independent Set Problem
  • Fire Station Problem (set covering)
  • 0-1 Integer programming
  • Integer Programming
  • Project management with resource constraints
  • and thousands more

82
Okay Should we give up?
NO WAY!!!
  • Here is why
  • The theory of NP-completeness is only a worst
    case result. Not all problem instances are as
    hard as the worst case.
  • Real problems tend to have sub-problems that are
    tractable and by exploiting the structure of such
    sub-problems using efficient algorithms such Unit
    Propagation, LP, Min Cost Flow, Transportation.
    Assignment and shortest path methods we can solve
    much larger problem instances.
  • In the 1970s we could only solve Binary Integer
    Programming instances with ? 100 variables. By
    exploiting the structure we can now solve real
    world instances with over 120,000 variables and
    4000 functional constraints.
  • In the 1990s we could only solve Satisfiability
    instances with ? 50 variables and 200 clauses.
    By using randomization and learning to exploit
    the structure we can now solve real world
    instances with over 1,000,000 variables and
    5,000,000 functional clauses.

83
NP-Complete and NP-Hard Problems
Planning and Scheduling And Supply Chain
Management
Data Analysis Data Mining
Protein Folding And Medical Applications
Capital Budgeting And Financial Appl.
Information Retrieval
Combinatorial Auctions
Software Hardware Verification
Many more applications!!!
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