1. CSE 550 Combinatorial Algorithms and Intractability
Instructor: Arun Sen
Office: BYENG 530
Tel:
Office Hours: MW 3:30-4:30 or by appointment

3. Two additional (recommended) books
Approximation Algorithms for NP-hard Problems – Dorit S. Hochbaum
Approximation Algorithms – Vijay V. Vazirani

4. Grading Policy
There will be one mid-term and a final. In addition, there will be programming and homework assignments
Mid-term 30%
Final 40%
Assignments 30%
For 90% will ensure A, 80% will ensure B, 70% will ensure C and so on
Loss of points due to late submission of assignments
1 day 50%
2 days 75%
3 days 100%

5. What is a Combinatorial Algorithm?
A combinatorial algorithm is an algorithm for a combinatorial problem.
What is a Combinatorial Problem?
Combinatorics is the branch of mathematics concerned with the study of arrangements, patterns, designs, assignments, schedules, connections and configurations.
Examples
A shop supervisor prepares assignments of workers to tools or work areas
An Industrial Engineer considers production schedules and workplace configurations to maximize production
A geneticist considers arrangements of bases into chains of DNA and RNA

6. Types of Combinatorial Problems
Three types of problems in Combinatorics
Existence Problems
Counting Problems
Optimization Problems
Optimization Problems are concerned with the choice of the “best” (according to some criterion) solution among all possible solutions.
In this class, we will focus on optimization and related problems.

7. How many binary trees can you draw with n nodes?
n=1: b1=1
n=2: b2=2
n=3: b3=5

8. How many binary trees can you draw with n nodes?
n= b4=14

9. How many binary trees can you draw with n nodes
How many binary trees can you draw with n nodes? Suppose b(n) is the number of binary trees that can be constructed with n nodes. In that case, b(n) can be expressed with the following recurrence relation

10. Combinatorial Problem in Manufacturing
Various wafers (tasks) are to be processed in a series of stations. The processing time of the wafers in different stations is different. Once a wafer is processed on a station it needs to be processed on the next station immediately, i.e., there cannot be any wait. In what order should the wafers be supplied to the assembly line so that the completion time of processing of all wafers is minimized?

11. S S S8 w1
t11
t12
t18 w2
t21
t22
t28 w3
t31
t32
t38

12. Completion Time in the first ordering = 13
w1 : t11 = 4, t12= 5;
w2 : t21 = 2, t22 = 4;
w1 :
w2 :
w2:
w1:
Completion Time in the first ordering = 13
Completion Time in the second ordering = 11
S1 : 4
S2 : 5
S1: 2
S2 : 4
S1:2
S2 : 4
S1: 2
S2 : 4
S1 : 4
S2 : 5

13. Sensor Placement Problem
Sensor Placement in a Temperature Sensitive Environment

14. Sensor Placement Problem
Sensor Placement in a Temperature Sensitive Environment

15. Sensor Placement Problem
Bio-sensors implanted in human body dissipate energy during their operation and consequently raise the temperature of the surrounding.
A temperature sensitive environment like the human body or brain can tolerate increase in temperature only up to a certain threshold.
One needs to make sure that the rise in temperature due to the operation of implanted bio-sensors in temperature sensitive environments such as human/animal body does not exceed the threshold and cause any adverse impact.

16. Thermal Model and Analysis
A sensor is expected to operate for a certain duration of time.
The rise is temperature in the area surrounding the sensor will be dependent on the duration of operation.
The sensor surroundings will attain a maximum temperature during the time of operation (steady state temperature).
If the steady state temperature of a sensor exceeds the maximum allowable threshold in the surrounding, such a sensor cannot be deployed.
Question: Is it possible that a sensor whose steady state temperature does not exceed the threshold when operating in isolation, may exceed the threshold when operating with multiple other sensors?

17. Thermal Model and Analysis
We perform a thorough analysis of the heat distribution phenomenon in a temperature sensitive environment and come to the following conclusion:
There exists a critical inter-sensor distance dcr, such that if the distance between any two deployed sensors is less than dcr, then the temperature in the vicinity of the sensors will exceed the maximum allowable threshold.
Therefore, attention must be paid during sensor deployment to ensure that the distance between any two sensors is at least as large as dcr.

18. Sensor Coverage Problem
Given:
A set of locations (or points pi) to be sensed
A set of potential locations (or points q i ) for the placement of the sensors
A minimum separation distance (dcr) between each pair of sensors.
Objective:
To deploy as few sensors as possible in the potential placement locations such that all points pi are sensed and the distance between any two sensors is at least as large as dcr.
Assumption
Each sensor is capable of sensing a circular area of radius rsen with the location of the sensor being the center of the circle.

19. Sensor Coverage Problem

20. Sensor Coverage Problem
Formal definition:

21. Set Cover Problem

22. Sensor Coverage as Generalized Set Cover Problem

23. Search Space The solution is somewhere here
Solution can be found by exhaustive search in the search space
Search space for the solution may be very large
Large search space implies long computation time to find solution (?)
Not necessarily true
Search space for the sorting problem is very large
The trick in the design of efficient algorithms lies in finding ways to reduce the search space

24. Evaluating Quality of Algorithms
Often there are several different ways to solve a problem, i.e., there are several different algorithms to solve a problem
What is the “best” way to solve a problem?
What is the “best” algorithm?
How do you measure the “goodness” of an algorithm?
What metric(s) should be used to measure the “goodness” of an algorithm?
Time
Space ** What about Power?

25. Problem and Instance Algorithms are designed to solve problems
What is a problem?
A problem is a general question to be answered, usually processing several parameters, or free variables, whose values are left unspecified. A problem is described by giving (i) a general description of all its parameters and (ii) a statement of what properties the answer, or the solution, required to satisfy.
What is an instance?
An instance of a problem is obtained by specifying particular values for all the problem parameters.

26. Traveling Salesman Problem
Instance: A finite set C={c1, c2, …, cm} of cities, a distance d(ci, cj) є Z+ for each pair of cities ci, cj є C and a bound B є Z+ (where Z+ denotes the positive integers).
Question: Is there a tour of all cities in C having total length no more than B, that is an ordering <cπ(1), cπ(2), …, cπ(m)> of C such that,

27. Algorithms are general step-by-step procedures for solving problems.
An algorithm is said to solve a problem Π if that algorithm can be applied to any instance I of Π and is guaranteed always to produce a solution for that instance I.
In general we are interested in finding the most efficient algorithm for solving a problem.
The time requirements of an algorithm are expressed in terms of a single variable, the size of a problem instance, which is intended to reflect the amount of input data needed to describe the instance.

28. Measuring efficiency of algorithms
One possible way to measure efficiency may be to note the execution time on some machine
Suppose that the problem P can be solved by two different algorithms A1 and A2.
Algorithms A1 and A2 were coded and using a data set D, the programs were executed on some machine M
A1 and A2 took 10 and 15 seconds to run to completion
Can we now say that A1 is more efficient that A2?

29. Measuring efficiency of algorithms
What happens if instead of data set D we use a different dataset D’?
A1 may end up taking more time than A2
What happens if instead of machine M we use a different machine M’?
If one want to make a statement about the efficiency of two algorithms based on timing values, it should read “A1 is more efficient that A2 on machine M, using data set D”, instead of an unqualified statement like “A1 is more efficient that A2”

30. Measuring efficiency of algorithms
The qualified statement “A1 is more efficient that A2 on machine M, using data set D” is of limited value as someone may use different data set or a different machine
Ideally, one would like to make an unqualified statement like “A1 is more efficient that A2” , that is independent of data set and machine
We cannot make such an unqualified statement by observing execution time on a machine
Data and Machine independent statement can be made if we note the number of “basic operations” needed by the algorithms
The “basic” or “elementary” operations are operations of the form addition, multiplication, comparison etc

31. Analysis of Algorithms
Size= n
Time
Compl Func 10 20 30 40 50 60
(A1) n
sec
(A2) n2
(A3) n3
(A4) n5
(A5) 2n
(A6) 3n
.00002
sec
.00003
sec
.00004
sec
.00005
sec
.00006
sec
.0001
sec
.0004
sec
.0009
sec
.0016
sec
.0025
sec
.0036
sec
.001
sec
sec sec sec sec sec
.1 sec
3.2 sec sec min min min
.001
sec
1.0
sec
17.9
min
12.7
days
35.7
years
366 centuries
.059
sec
* *1013
min years cents cents cents.

32. Size of Largest Problem Instance Solvable in 1 Hour
Time complexity function
With present computer
With computers 100 times faster
With computer 1000 times faster n N1 n2 N2 n3 N3 n5 N4 2n N5 3n N6
100 N1
1000 N1
10 N2
31.6 N2
4.64 N N3
2.5 N N4 N N
N N

33. Growth of Functions: Asymptotic Notations
O(g(n)) = {f(n): there exists positive constants c and n0 such that 0<=f(n)<=c * g(n) for all n >= n0}
Ω(g(n)) = {f(n): there exists positive constants c and n0 such that 0<=c * g(n)<=f(n) for all n >= n0}
Q(g(n)) = {f(n): there exists positive constants c1, c2 and n0 such that 0<= c1 * g(n)<=f(n)<=c2*g(n) for all n >= n0}
o(g(n) = {f(n): for any positive constant c>0 there exists a constant n0>0 such that 0<=f(n)<c * g(n) for all n >= n0}
w(g(n)) = {f(n): for any positive constant c>0 there exists a constant n0 such that 0<=<c * g(n)< f(n) for all n >= n0}
A function f(n) is said to be of the order of another function g(n) and
is denoted by O(g(n)) if there exists positive constants c and n0 such
that 0<=f(n)<=c * g(n) for all n >= n0}

34. Basic Operations and Data Set
To evaluate efficiency of an algorithm, we decided to count the number of basic operations performed by the algorithm
This is usually expressed as a function of the input data size
The number of basic operations in an algorithm
Is it independent of the data set ?
Is it dependent on the data set?

35. Given a set of records R1, …, Rn with keys k1, …,kn
Given a set of records R1, …, Rn with keys k1, …,kn. Sort the records in ascending order of the keys.

36. Basic Operations and Data Set
The number of basic operations in an algorithm
Is it independent of the data set ?
Is it dependent on the data set?
If the number of basic operations in an algorithm depends on the data set then one needs to consider
Best case complexity
Worst case complexity
Average case complexity
What does “average” mean?
Average over what?

37. Given n elements X[1], …, X[n], the algorithm finds m and j such that m = X[j] = max 1<=k<=n X[k], and for which j is as large as possible.
Algorithm FindMax
Step 1. Set j  n, k  n – 1, m  X[n]
Step 2. If k=0, the algorithm terminates.
Step 3. If X[k] <= m, go to step 5.
Step 4. Set j  k, m  X[k].
Step 5. Decrease k by 1, and return to step 2

38. Computational Speed-up and the Role of Algorithms
Moore’s law says that computing power (hardware speed) doubles every eighteen months
How long will it take to have a thousand-fold speed-up in computation, if we rely on hardware speed alone?
Answer: 15 years
Expected cost: significant
How long will it take to have a thousand-fold speed-up in computation, if we rely on the design of clever algorithms?
Thousand-fold speed-up can be attained if currently used O(n5) complexity algorithm is replaced by a new algorithm with complexity O(n2) for n=10.
How long will it take to develop a O(n2) complexity algorithm which does the same thing as the currently used O(n5) complexity algorithm?
Answer: May be as little as one afternoon
Ingredients needed
Pencil
Paper
A beautiful mind
Expected cost: significantly less than what will be needed if we rely on hardware alone

39. Computational Speed-up and the Role of Algorithms
A clever algorithm can achieve overnight what progress in hardware would require decades to accomplish.
“The algorithm things are really startling, because when you get those right you can jump three orders of magnitude in one afternoon.”
William Pulleyblank
Senior Scientist, IBM Research

40. Algorithm Design Techniques
Divide and Conquer
Dynamic Programming
Greedy Algorithms
Backtracking
Branch and Bound
Approximation Algorithms
Probabilistic (Randomized) Algorithms
Mathematical Programming
Parallel and Distributed Algorithms
Simulated Annealing
Genetic Algorithms
Tabu Search

41. How do you “prove” a problem to be “difficult”?
Suppose that the algorithm you developed for the problem to be solved (after many sleepless nights) turned out to be very time consuming
Possibilities
You haven’t designed an efficient algorithm for the problem
May be you are not that great an algorithm designer
May be you are a better fashion designer
May be you have not taken CSE 450/598
May be the problem is difficult and more efficient algorithm cannot be designed
How do you know that more efficient algorithm cannot be designed?
It is difficult to substantiate a claim that more efficient algorithm cannot be designed
Your inability to design an efficient algorithm does not necessarily mean that the problem is “difficult”
It may be easier to claim that the problem “probably” is “difficult”
How do you substantiate the claim that the problem “probably” is “difficult”?
What if you line up a bunch of “smart” people who will testify that they also think that the problem is difficult?
Theory of NP-Completeness

42. Taxonomy of Problems Problems Undecidable Decidable Intractable
(deterministically)
Tractable
(deterministically)
Tractable
(non-deterministically)
Intractable
(non-deterministically)
NP-Complete Problems
(Most likely deterministically
Intractable

43. Complexity of Algorithms and Problems
In algorithms classes (e.g., CSE 450) we make distinctions between algorithms of complexity O(n2), O(n3), and O(n5).
In this class, we take a much coarse grain view and divide algorithms into only two classes – polynomial time algorithms and non-polynomial time algorithms.
Polynomial time algorithms – Good
Non-polynomial time algorithms - Bad

44. Good vs. Bad (in Algorithms)
Exponential time algorithms should not be considered “good” algorithms.
Most exponential time algorithms are merely variations of exhaustive search.
Polynomial time algorithms generally are made possible only through gain of some deeper insight into the structure of a problem.

45. Easy and Difficult Problems
A problem is easy if a polynomial time algorithm is known for it.
A problem may be suspected to be difficult if
a polynomial time algorithm cannot be developed for it, even after significant time and effort.

46. Theory of NP-Completeness
Complexity of an algorithm for a problem says more about the algorithm and less about the problem
If a low complexity algorithm can be found for the solution of a problem, we can say that the problem is not difficult
If we are unable to find a low complexity algorithm for the solution of a problem, can we say that the problem is difficult?
Answer: No
NP-Completeness of a problem says something about the problem
Problems may or may not be NP-Complete – not the algorithms

47. Problems and Algorithms for their solution
Problem P
Algorithm 3
Complexity: O(2n)
Algorithm 1
Complexity: O(n)
Algorithm 2
Complexity: O(n4)

48. Complexity of a Problem

51. How to prove a problem difficult?
Is the approach of lining up a group of famous people really going to work?
Answer: Probably not
Why would a group of famous people be interested in working on your problem?
“If the mountain does not come to Mohammed, Mohammed goes to the mountain”
If the famous people are not interested in working on your problem, you transform their problem into yours.
If such a transformation is possible, you can now claim that if your problem can easily be solved, so can be theirs.
In other words, if their problem is difficult, so is yours.

52. Problem Transformation – Hamiltonian Cycle Problem
A cycle in a graph G = (V, E) is a sequence <v1, v2, …, vk> of distinct vertices of V such that {vi, vi+1} e E for 1 <= i < k and such that {vk, v1} e E.
A Hamiltonian cycle in G is a simple cycle that includes all the vertices of G.
Hamiltonian Cycle Problem
Instance: A graph G = (V, E)
Question: Does G contain a Hamiltonian cycle?

53. Traveling Salesman Problem
Instance: A finite set C={c1, c2, …, cm} of cities, a distance d(ci, cj) є Z+ for each pair of cities ci, cj є C and a bound B є Z+ (where Z+ denotes the positive integers).
Question: Is there a tour of all cities in C having total length no more than B, that is an ordering <cπ(1), cπ(2), …, cπ(m)> of C such that,

54. No-wait Flow-shop Scheduling Problem
S S S8 w1
t11
t12
t18 w2
t21
t22
t28 w3
t31
t32
t38

55. Problem Transformation
No-wait Flow-shop Scheduling Problem can be transformed into Traveling Salesman Problem
How?
We will see it later
Hamiltonian Cycle problem can be transformed to Traveling Salesman Problem
From an instance of the HC Problem, the graph G = (V, E), (|V| = n), construct an instance of the TSP problem as follows: Construct a completely connected graph G’ = (V’, E’) where (|V’| = |V|). Associate a distance with each edge of E’. For each edge e’ e E’, if e’ e E then dist(e’) = 1, otherwise dist(e’) = 2. Set B, a problem parameter of the TSP problem, equal to n.

56. Problem Transformation
Claim: Graph G contains a Hamiltonian Cycle, if and only if there is a tour of all the cities in G’, that has a total length no more than B.
If G has a HC <v1, v2, …, vn>, then G’ has a TSP tour of length n = B, because each intercity distance traveled in the tour corresponds to an edge in G and hence has length 1.
If G’ has a TSP tour of length n = B, then each edge e that contributes to the tour must have dist(e) = 1 (because the tour is made up of n edges). It implies that these edges are present in G as well. These set of edges makes up a Hamiltonian Cycle in G.

57. Coping with NP-Complete Problems
A combinatorial optimization problem P is either a minimization or a maximization problem and consists of the following three parts
A set DP of instances
For each instance I a finite set SP(I) of candidate solutions for I
A function mP that assigns to each instance I and each candidate solution s є SP(I) a positive rational number mP(I, s) called the solution value for s.
If a combinatorial optimization problem is NP-Complete,
it might take an absurdly long time (e.g., 300 centuries) to find the optimal solution for the problem.
Probably cannot wait 300 centuries to find the solution
However, the problem does not go away. One still has to find a solution!

58. Approximation Algorithms
If the optimal solution is unattainable then it is reasonable to sacrifice optimality and settle for a “good” (close to optimal) feasible solution that can be computed efficiently (i.e., within some reasonable amount of time).
We would like to sacrifice as little optimality as possible, while gaining as much as possible in efficiency.
Trading-off optimality in favor of tractability is the paradigm of approximation algorithms
- Dorit. S. Hochbaum
Approximation Algorithms for NP-Hard Problems

59. “Cost” vs. “Quality” of a solution
“Cost” of a solution: Time spent in finding a solution
“Quality” of a solution: Closeness of the solution to the optimal
Tradeoff between “thinking time” vs. “execution time”
Thinking Time
Execution Time
Time

60. Approximation Algorithms
Let  be an optimization (minimization or maximization) problem and A an algorithm which, given an instance I of , returns a feasible (but not necessarily optimal) solution denoted by APP(I). Let the optimal solution be denoted by OPT(I). The algorithm A is called an -approximation algorithm for  for some  ≥ 0 if and only if
| APP(I) – OPT(I) | / OPT(I) ≤  for all instances I

61. Yet Another Job Scheduling Problem
P1, …, Pm: A set of m independent processors with similar (dissimilar) performance characteristics
T1, …, Tn: A set of n independent tasks with no ordering relationship between them
If the processors are dissimilar (heterogeneous computing environment), the execution time of a task on different processors are different.
tij = Execution time of task Ti on Processor Pj
If the processors are similar (homogeneous computing environment), the execution time of a task on different processors are same.

62. Yet Another Job Scheduling Problem
Total execution time used by Processor Pj is the sum of the execution times of the tasks assigned to this processor
Makespan of an assignment is maximum of the total execution times of individual processors
Objective: Find the assignment that minimizes the makespan
Question: How difficult is it to find the schedule with the minimum makespan?

63. Algorithm for the Job Scheduling Problem (Heterogeneous Environment)
Step1: for j:=1 to m do
Fj := 0;
Step2: for i:=1 to n do
begin
Step Find k where k is the smallest integer j for which Fj + ti, j is minimum (1≤j≤ m)
Step Assign task Ti to processor Pk
Step Update Fk  Fk + ti, j
end

64. Jobs
Execution
Time
Case 1
Case 2

65. Case 1:

66. or as
and

67. Case 2
F4 was the finish time at the end of scheduling jobs J1,.., J7. Let the finish time on the processors at this time be (before scheduling job J7).

68. or

69. Steiner Tree Problem
Instance: Graph G (V, E) , a weight w(e) є Z0+ for each edge e є E, a subset R V, and a positive integer bound B.
Question: Is there a subtree of G that includes all the vertices of R such that the sum of the weights of the edges in the subtree is no more than B.

70. Heuristic Algorithm for Steiner Trees(H)
INPUT: An undirected graph G = (V, E) and a set of Steiner Points S  V.
OUTPUT : A Steiner Tree TH for G and S.
Step 1: Construct the complete undirected graph G1 =(V1, E1) from G and S., in such a
Way that V1 = S, and every {vi, vj}  E1, the weight (length/cost) on the edge {vi, vj}
is equal to the length of the shortest path from vi to vj.
Step 2: Find the minimal spanning tree T1 of G1. (If there are several minimal spanning
trees, pick an arbitrary one.
Step 3: Construct the sub graph Gs of G by replacing each edge in T1 by its
corresponding shortest path in G. (If there are several shortest paths, pick an arbitrary
one.)
Step 4: Find the minimal spanning tree Ts of Gs. (If there are several minimal spanning
trees, pick an arbitrary one.)
Step 5: Construct a Steiner Tree TH, from T by deleting edges in T, if necessary, so that
all the leaves in TH are Steiner Points.

71. Steiner Points: {v1, v2, v3, v4}10 1
1/2 8 2 9 v1 v2 v7 v8 v9 v6 v5 v4 v3
(b) 4 v1 v3 v2 v4
(a)
(c)

72. 1
1/2 2 v1 v2 v7 v8 v9 v6 v5 v4 v3 1
1/2 2 v1 v2 v7 v8 v9 v6 v5 v4 v3 1 2 v1 v2 v9 v6 v5 v4 v3
(d)
(e)
(f)

73. Analysis of the Algorithm
G = (V, E): Input Graph, S: Steiner Points (|S| = k), H: Heuristic Algorithm
DH = Total length on the edges of the Steiner Tree TH, produce by the algorithm H.
DMIN = Total length on the edges of the minimal Steiner Tree TMIN.
m = Total number of leaves in TMIN.
Construct a loop L around TMIN that traverses each edge of TMIN exactly two times.
Every leaf in TMIN appears exactly once in L.
If ui, uj are two “consecutive” leaves in the loop, then the subpath connecting ui to
uj is a simple path.
We may regard the loop L as composed of m simple subpaths, each connecting a
leaf to another leaf.
Construct a path P by deleting the longest “leaf to leaf” subpath from L.
Length(P) ≤ (1 – 1/m) Length(L)

74. DH / DMIN ≤ 2 ( 1 – 1/m ) Performance Bound is 2.
Every edge in TMIN appears at least once in P.
Let (w1. w2, …, wk) be the k distinct Steiner points appearing in P, in that order.
Length(P) ≤ (1 – 1/m) Length(L) = 2 (1 – 1/m) Length (TMIN) = 2 (1 – 1/m) DMIN
Length(P) ≥ Length of a spanning tree for G1 consisting of the edges
{w1, w2 }, { w2, w3}, …., { wk-1, wk}
Length(P) ≥ Length of the minimal spanning tree for G1.
Length(P) ≥ DH
Length(P) ≤ 2 ( 1- 1/m) DMIN
DH ≤ 2 ( 1 – 1/m ) DMIN
DH / DMIN ≤ 2 ( 1 – 1/m ) Performance Bound is 2.

75. Approximation Ratios in Graphs
2-approximation [3 independent papers, ]
Last decade of the second millennium:
11/6 = [Zelikovsky]
16/9 = [Berman & Ramayer]
PTAS with the limit ratios:
1.73 [Borchers & Du]
1+ln2 = [Zelikovsky]
5/3 = [Promel & Steger]
1.64 [Karpinski & Zelikovsky]
1.59 [Hougardy & Promel]
2000:
1.55 = ln3 / 2
Cannot be approximated better than 1.004
In the Moving-Target TSP formulation, there are a set of targets such that each target has an initial velocity and starting position.
A pursuer must find a tour which intercepts all of the targets in the minimum amount of time.
In this work, we also consider variations of the Moving-Target TSP formulation. These variations correspond to the a Moving-Target variant of the traditional Vehicle Routing Problem, where vehicles with fixed capacity must supply customers with fixed demand.
Vehicles may resupply at a central depot in order to service more customers. In our formulation, though, the customers, or targets, may be moving.

76. The description of a problem instance that we provide as input to the computer can be viewed as a single finite string of symbols chosen from a finite input alphabet.
Each problem has associated with it a fixed encoding scheme which maps problems into the strings describing them. The input length for an instance I of a problem Π is defined to the number of symbols in the description of I obtained from the encoding scheme for Π.

77. The time complexity function for an algorithm expresses its time requirements by giving, for each possible input length, the largest amount of time needed by the algorithm to solve a problem instance of that size.
A function f(n) is said to be O(g(n)) whenever there exists constant c and n0 such that |f(n)| ≤ c*|g(n)| for all n> n0. A polynomial time algorithm is defined to be one whose time complexity function is O(p(n)) for some polynomial function p, where n is used to denote the input length. Any algorithm whose time complexity function cannot be so bounded is called as exponential time algorithm.

78. A problem is intractable if it is so hard that no polynomial time algorithm can possibly solve it.
Time complexity as defined is a worst-case measure, and the fact that an algorithm has time complexity 2n means that at least one problem instance of size n requires that much time.
The intractability of a problem turns out to be independent of the particular encoding scheme and computer model used for determining time complexity.

79. “Reasonable” encoding scheme: Although we do not formalize the notion of “reasonableness”, the following two conditions capture much of the notion:
the encoding of an instance I should be concise and not “padded” with unnecessary information or symbols, and
Numbers occurring in I should be represented in binary (or any fixed base other than 1).

80. Two different causes of intractability
The problem is so difficult that exponential amount of time is needed to discover a solution.
The solution itself is required to be so extensive that it cannot be described with an expression having length bounded by a polynomial function of the input length.
Undecidable Problem: No algorithm can be developed for solving it.

81. “It is impossible to specify any algorithm which, given an arbitrary computer program and an arbitrary input to that program, can decide whether or not the program will eventually halt when applied to that input.” [Alan Turing, 1936]
Decidable Intractable Problem: Such problems cannot be solved in polynomial time using even a “nondeterministic” computer model, which has the ability to pursue an unbounded number of independent computational sequences in parallel.

82. All the provably intractable problems known to date are either undecidable or “non-deterministically” intractable.
However, most of the apparently intractable problems encountered in practice are decidable and can be solved in polynomial time with the aid of non-deterministic computer.
Problem Reduction: The technique used to demonstrate that two problems are related is that of reducing one to the other, by giving a constructive transformation that maps any instance of the first problem into an instance of the second.

83. Decision Problem: problem with only yes/ no answer.
A decision problem Π consists simply of a set DΠ of instances and a subset YΠ, subset of DΠ of yes instances.
Decision problems and optimization problems: So long as the cost function is relatively easy to evaluate the decision problem can be no harder than the corresponding optimization problem. (many decision problems including TSP, can be shown to be “no easier” than the corresponding optimization problem)

84. The reason that the study of the theory of NP-completeness is restricted to decision problems is that they have a very natural, formal counterpart, which is a suitable object to study in mathematically precise theory of computation known as “language”.
The correspondence between decision problems and language is brought about by the encoding scheme we use for specifying the instances.