Algorithm Design | Approximation Algorithm | Traveling Salesman Problem with Triangle Inequality
Summary
TLDRThis video script delves into the Traveling Salesman Problem (TSP), focusing on an approximation algorithm technique using the triangular inequality. It explains the TSP, where a salesman must visit a set of cities and return to the origin, and introduces the concept of Hamiltonian cycles. The script discusses the complexity of TSP, which is part of NP-complete problems, and outlines an approach to simplify the problem using the triangular inequality to create an approximation algorithm. The method involves creating a minimum spanning tree (MST) using Prim's algorithm, then refining the solution by applying the triangular inequality to reduce the cost. The script also covers the theoretical aspect, explaining how the approximation algorithm provides a solution that is at most twice the optimal cost, hence a 2-approximation algorithm. The time complexity of the algorithm is polynomial, making it a feasible approach to solving TSP.
Takeaways
- 🧳 The video discusses the Traveling Salesman Problem (TSP) and its solution using an approximation algorithm based on the triangular inequality.
- 🔍 TSP is a well-known problem in computer science that is part of the NP-complete class, implying that finding an exact solution can be computationally intensive.
- 📚 The approximation algorithm uses the concept of triangular inequality to simplify the problem and make it more approachable in polynomial time.
- 📈 The script introduces the concept of a Hamiltonian cycle, which is a closed loop on a graph that visits each vertex exactly once and returns to the starting vertex.
- 🌐 The algorithm starts by selecting any vertex as a root and then constructs a minimum spanning tree (MST) using Prim's algorithm, which is efficient and suitable for this purpose.
- 📝 The definition of the cost of a cycle in TSP is given as the sum of the costs of all edges within the cycle, and the goal is to minimize this cost.
- 📉 The triangular inequality states that for any three vertices u, v, and w, the direct cost from u to v should be less than or equal to the cost of going from u to w and then w to v.
- 🔄 The script explains that by applying the triangular inequality, the algorithm iteratively removes edges that do not contribute to the optimal solution, thus approximating the TSP.
- 📊 The video references a theorem that the approximation algorithm provides a solution that is at most twice the cost of the optimal solution (2-approximation).
- 🕒 The time complexity of the algorithm is polynomial, specifically O(e log V) or O(V + e log V), where e is the number of edges and V is the number of vertices, making it computationally feasible.
Q & A
What is the Traveling Salesman Problem (TSP)?
-The Traveling Salesman Problem (TSP) is an algorithmic problem where a salesman is given a list of cities and must determine the shortest possible route that allows him to visit each city once and return to his original city. It is a well-known problem in the field of combinatorial optimization and is known to be NP-hard.
Why is the TSP considered a part of NP complete or NP hard?
-The TSP is considered NP hard because it is a problem whose solutions can be verified in polynomial time, but finding the solution is computationally intensive and requires exponential time in the worst case, making it infeasible to solve for large instances.
What is a Hamiltonian cycle and how is it related to TSP?
-A Hamiltonian cycle is a closed loop on a graph where every node (city, in the context of TSP) is visited exactly once. It is related to TSP because the goal of TSP is to find the shortest Hamiltonian cycle that visits all given cities and returns to the starting city.
What is the purpose of using the triangular inequality in the TSP approximation algorithm?
-The triangular inequality is used to simplify the problem and make it easier to solve in polynomial time. It helps in creating an approximation algorithm for TSP by ensuring that the direct distance between two points is less than or equal to the sum of the distances through any third point, thus avoiding inefficient routes.
How does the approximation algorithm using triangular inequality work in the context of TSP?
-The approximation algorithm starts by selecting any vertex as a root and then creating a minimum spanning tree (MST) from that vertex. After the MST is formed, a Hamiltonian cycle is constructed by visiting each vertex in pre-order and applying the triangular inequality to ensure the shortest paths are chosen, resulting in an approximate solution to the TSP.
What is the significance of the minimum spanning tree (MST) in the approximation algorithm for TSP?
-The minimum spanning tree is significant because it connects all the vertices with the minimum possible total edge weight. It serves as a foundation for the approximation algorithm by providing a starting point for constructing a Hamiltonian cycle that aims to be as close to the optimal TSP solution as possible.
What does the term 'full work' refer to in the context of the TSP approximation algorithm?
-In the context of the TSP approximation algorithm, 'full work' refers to the process of traversing all the vertices in a pre-ordered manner, effectively creating a cycle that visits each vertex twice. This is used to apply the triangular inequality and find a more optimal path.
What is the meaning of a 'two-approximation algorithm' for TSP?
-A 'two-approximation algorithm' for TSP is an algorithm that provides a solution that is at most twice the cost of the optimal solution. It means the algorithm guarantees that the path length it finds will not be more than twice as long as the shortest possible path.
How is the time complexity of Prim's algorithm relevant to the TSP approximation algorithm?
-The time complexity of Prim's algorithm, which is used to create the minimum spanning tree in the TSP approximation algorithm, is relevant because it indicates the efficiency of the algorithm. Prim's algorithm has a time complexity of O(E log V) or O(V^2), making it a polynomial time algorithm suitable for the approximation approach.
What are the naming conventions used for different components of the TSP approximation algorithm as described in the script?
-The naming conventions used in the script are as follows: 'H*' denotes the optimal tour, 'CT' represents the cost of the minimum spanning tree, 'CW' is the cost of the full work, and 'CH' represents the approximate cost of the Hamiltonian cycle obtained after applying the triangular inequality.
Outlines
🧳 Introduction to the Traveling Salesman Problem and Approximation Algorithm
The video introduces the Traveling Salesman Problem (TSP), a classic algorithmic problem in the field of computer science. The speaker discusses using an approximation algorithm to simplify the TSP, which is known to be NP-hard. The approximation algorithm leverages the triangular inequality to find a solution in polynomial time. The concept of a Hamiltonian cycle, a closed loop that visits each vertex once and returns to the origin, is also explained as fundamental to solving the TSP.
📏 Applying Triangular Inequality in Approximation Algorithms
This paragraph delves into the specifics of using the triangular inequality to create an approximation algorithm for the TSP. The triangular inequality principle is explained, where the direct distance between two points (U to V) is compared with the sum of the distances via an intermediary point (U to W and W to V). The algorithm selects the shorter path, which aids in finding a near-optimal solution for the TSP. The paragraph also references Cormen's 'Introduction to Algorithms' as a resource for further understanding.
🌳 Constructing a Minimum Spanning Tree for TSP Approximation
The speaker outlines the process of creating a minimum spanning tree (MST) as part of the approximation algorithm for TSP. Starting with an arbitrary vertex as the root, the algorithm extends the tree by adding edges, ensuring the total weight is minimized. Two algorithms for MST, Prim's and Kruskal's, are mentioned, with Prim's algorithm being chosen for its sequential edge addition, which aligns with the step-by-step nature of the approximation process.
🔄 Full Work and Triangular Inequality in TSP
The concept of 'full work' is introduced, which involves visiting each vertex in a pre-order sequence, effectively creating a cycle. This cycle is then optimized using the triangular inequality to remove less efficient paths, thus improving the approximation's accuracy. The paragraph emphasizes the iterative nature of this process, where the full work is refined to better approximate the optimal TSP solution.
📊 Theoretical Foundation of the 2-Approximation Algorithm
The video script provides a detailed explanation of the theoretical underpinnings of the 2-approximation algorithm for TSP. It presents mathematical models and proofs to demonstrate that the cost of the approximate solution is at most twice the cost of the optimal solution. The notation used in the script, such as H* for the optimal tour and CT for the minimum spanning tree, helps to formalize the proof and establish the efficiency of the approximation method.
🔢 Time Complexity and Conclusion of the Approximation Method
The final paragraph wraps up the discussion by emphasizing the polynomial time complexity of the approximation algorithm, which is crucial for its practical application. The speaker mentions the time complexity of Prim's algorithm as O(e log V), where e is the number of edges, and V is the number of vertices. The use of triangular inequality is highlighted as a key factor in achieving this polynomial time complexity, concluding the explanation of the approximation method for the TSP.
Mindmap
Keywords
💡Traveling Salesman Problem (TSP)
💡Approximation Algorithm
💡Triangular Inequality
💡NP-Complete
💡Hamiltonian Cycle
💡Minimum Spanning Tree (MST)
💡Prim's Algorithm
💡Pre-order Traversal
💡Full Work
💡Time Complexity
Highlights
Introduction to the Traveling Salesman Problem (TSP) with triangular inequality using approximation algorithm technique.
Explanation of approximation algorithm and its application to simplify TSP to a polynomial time problem.
TSP's classification as part of NP-complete, highlighting its computational complexity.
Overview of Hamiltonian cycle concept and its relevance to TSP.
Detailed description of the TSP, emphasizing the need to return to the starting city after visiting all others.
Introduction of triangular inequality as a method to create an approximation algorithm for TSP.
Example of applying triangular inequality to choose the most cost-effective path in TSP.
The selection process for any vertex as a starting point in the approximation algorithm.
Creation of a minimum spanning tree (MST) as a step in the approximation algorithm using Prim's algorithm.
Explanation of the list of vertices and their order based on pre-order traversal.
Concept of full work in TSP to ensure all vertices are visited according to triangular inequality.
Process of completing the triangle inequality to approximate the TSP solution.
Real-life example of approximation versus optimal solution in delivery scenarios.
Theorem and proof of the 2-approximation algorithm for TSP using triangular inequality.
Mathematical models and expressions used to prove the approximation algorithm's effectiveness.
Discussion on the time complexity of Prim's algorithm and its polynomial nature.
Final summary of the approximation methodology using triangular inequality for TSP.
Transcripts
hello everyone welcome to edil in this
video we are going to discuss about
traveling salesman problem with
triangular inequality using
approximation algorithm technique okay
the approximation basically we are using
is using triangular unquality how to
find the traveling salesman problem to
be easier towards some polinomial time
how do we expect okay because traveling
salesman is a part of NP complete or
somewh you can find it out in NP hard
also so but in general you can say it
lies between NP hard to NP complete but
maximum uh Network stops I found like
it's a part of NP complete okay so those
who are new kindly watch the previous
videos uh next we'll start with the
traveling salesman problem so your
traveling salesman problem is a problem
through which I must tell you like uh we
have already completed reduction in
detail so Hamilton Cy
we should introduce what is this suppose
you start with uh
vertex I'm just starting with a b c and
d okay suppose this is undirected graph
because in general whenever you say uh
traveling salesman problem so each
Vortex is considered as a CT okay so a
person is going to travel all the city
and when he or she started from a
position he or she should have to come
back to that particular position this is
what the traveling salesman problem is
so why we introduce hamiltonan cycle
this is I'm just repeating but this part
you can the full detail video is there
in reduction you can check that one and
you can understand like how the
traveling salesman is working fine okay
so for we introduced hamiltonan cycle or
hamiltonan circuit you can say suppose
you started from a you can reach B you
can reach C then you can reach D again
you can reach a this is one point or you
can start because though it is
undirectional undirected means it's a
bidirectional okay so you can start with
a to d d to C D to b or you can switch
to uh B to a clear there are two ways
you can definitely start from a and
start end at a also okay so for that
reason we are introducing hamiltonan
cycle okay or Hamilton circuit this is
what we have already discussed now our
main target is then why we use for
approximation the the basic question
okay so uh this is the definition as for
the definition which it is written like
for a suppose you are trying to you you
just converted a cycle like here it is
started and you just trying to convert
that one up to you know you try to count
this a this is what a is okay a cycle is
created now the cost cost if you want to
find out C stands for the cost which is
a sigma because all the uh ages
whichever you have connected like with
certain u and v with the vertices
whatever the edes you have connected or
might be within the cycle you just count
that one or you just some add that one
and this addition of all the cost of
Ages is called as the small that should
be minimum the
minimum it should be minimum okay
then only you can say this is your tsp
okay but this is sometime hard for us to
find out in case of reduction we have
set some rule here it is written and uh
if you are not restricted the tsp
restricted in the sense like if you have
not certain if have not framed any rules
for that one tsp is too hard to find out
okay so in in reduction we said all the
distance should be one then hamiltonian
cycle we run and we get it within a poal
time so this is what exactly we studied
like uh that is what we are trying to
find out the minimum with the problem
exactly what exactly we focused here we
are not keeping all the distance as one
or two or some same distance is not
there here the distance could be
different but we are introducing
triangular or triangle inequality okay
triangle inequality we are using through
which we are creating an approximation
algorithm for tsp to get the optimal
solution for it okay next triangle
inequality means for this example I've
just kept one example here suppose you
want to move from U to V there is one
way another way also this one clear so
how do you pick which one is good for
you like suppose u2v cost u2v means
direct uh you have two options like from
direct you have to check whether it is
less than Cost U to W and cost this cost
and this cost it represents okay so
whether it is equal less than equal to
this one or not if it is less than this
so we will pick UV instead of U to W and
W to V this is what the triangle
inequality if it is more if the distance
is more so definitely we'll pick this
one like U to V we will come and sorry U
to W then W to V we are reaching if you
will find it out by comparing this one
so definitely stop accepting this row
and we will pick this one CLE this is
what a triangle inequality it represents
it uh then using triangle uh inequality
can we reduce or can we give such
approximation algorithm or not that is
our aim and objective here clear so you
can check approximation algorithm we are
going to use so I'll start from this
book I'm following the book uh written
by Corman introduction to algorithm you
can check in the description also the
book and uh the textbook and the
reference book everything I have
mentioned there next uh what text are
select any vertex you this is not always
mandatory no you should start from the
end you can start from the bottom
nothing you you can start any vertex
select any vertex from the vertex given
okay and keep that was a root First Step
Second Step what it says we are just
creating a minimum spanning tree those
who are new kindly watch the full detail
video there in a minimum spinning tree
okay uh so here uh we kept the root R
based on that we created the minimum
sping tree as we have already discussed
whenever we studied the minimum spanning
tree there are two approaches or two
algorithms we discussed one is prims
algorithm other is cuscal algorithm so
here this is a PR algorithm we going to
use so because we are trying to find out
one by one with the ages that's why we
fixed up with the PES algorithm we
detail analysis will be done don't worry
about it first we will uh read each and
every sentence written there because
algorithm is a comprises of the
solutions that's why we first starting
with the solution then we are going
towards the problem
clear next H is a list of vertices what
exactly is our list of vertices you can
say ordered accordingly when they are
first visited in pre-order like whenever
we have already studied to visit a tree
there are three different ways one or
three different order we have studied
earlier in data structure one is
pre-order in order post order there are
three ways pre-order means we initially
we started visiting of the node for the
first first time in order means for the
second time post order means third time
clear we have already discussed in the
data structure video there it is clearly
mentioned so how to complete to the
pre-order first we have completed a
pre-order like H we will completely say
that one and return the hamiltonian
cycle hamiltonian cycle means we will
not repeat the same age again okay we
will say once I'll start with like the
the example I must tell you like we will
start from this end you can start your
movement but you will end up with this
one this is what your hamiltonian cycle
okay like you will definitely end at
this
position wherever you start you will
definitely end you can definitely end it
this position this is Hamilton cycle we
have discuss that that is what the
fourth line says clear so I hope you
understood what exactly it is written
now based on the example there in the
book I'm just going to tell you that one
this a represents this is a graph given
to us with uh you can say the weight are
different but here the weight is not
mentioned in the uh book that's why I'm
not you know adding that one so out of
this we are going to create MST minimum
spanning tree using prims why prims
because you can take uh the help of
crull also not an issue but they have
taken the PES okay so once the PES they
have collected like they started a as a
root from that one they just created an
MST okay next MST is a tree that we have
already discussed this C represents is a
full
work full
work full work means we are going to
just cover all all the vertices why
because we are trying to find out the
triangle inequality suppose you must say
like whether I if I'm coming to C to B
then I'm going towards C to H whether
this H is greater than this is or not if
this is greater than so I must have to
remove that one and I'll select this one
for for triangular inequality for
triangular
inequality or triangle inequality
I'm just writing in short for triangular
inequality we are actually doing the
full work okay next once it is done so
we have completed triangle inequality so
you can say this is your
approximate tsp traveling sales person
or traveling salesman problem
okay this is your approximate so instead
of B2 H C2 B then C2 H clear using
triangle
inequality this is your optimal
optimal or best you can say
tsp mean you know the problem optimal
how do you define in real life you know
that one is it will take at least 2
minutes to cover everything or 2 hours
to cover every CT is so you know that
this is optimal suppose you uh inform a
person to in a real life basically how
you will use that one optimal means the
experience fellow experience fellow
means I have already served the root
everything I know two hours is enough
but sometimes a person is coming I I I
hired him or her for delivery purposes
might be he or she will take like I'll
tell like which one that you should have
to cover first how you will complete
everything I instructed everything but
the new person he see tries to
understand tries to tackle all the paths
within less time might be it's near
optimal I told like 2 hours is enough
might be for him or her it took 3 hours
so this is what a real life example of
approximation with the optimal solution
optimal solution like here what they
said like A2 B then B2 H instead of H2 D
what they started h2f this is the
optimal answer okay which is represented
what format that we will discuss this is
how we have completed how it is working
clear now for the theorem for the
theorem how to prove it just check it in
general uh University type of exams this
type of questions could be come like how
to verify it is two approximation
algorithm clear so it is saying uh
approx tsp t is polom time to
approximation algorithm for traveling
salesman problem using triangle
inequality using triangle inequality we
are getting a two approximation means
two times it means 2
times
more
than
the
optimal value or optimal cost two time
more that that is what it represents two
approximation algorithm that we have
already discussed C
star C by C
star equal to row of n that row
represents two that is what we have
already discussed to
approximation then how to
prove this one try to understand now
clear so here H star denotes optimal
tour we have done the optimal tour H
star represents so this is our H star or
cost of H star you can say which is the
smallest for a timing because this is
our optimal one okay next T represents
the spanning tree which you have
completed for the first time it is
represented in the line two line two
means this one compute minimum spanning
tree whenever we have computed this one
is cost of spanning tree this is what CT
represents clear CT CH
star now for full work we'll decide full
work is represented as W clear so this
one is cost of w is represented as the
total cost or total weight covered by
the full work
okay
next this H represents which is obtained
by deleting the full work deleting the
vertices from the full work represents
the
approximate cost of
H
okay these are just the naming
conventions first we'll try to
understand what it does whenever we say
optimal one optimal is always greater
than the spanning tree why here you can
say whenever you are creating the tree
clear this is for one time one Ed should
be connected if there is a cycle
definitely I must tell you this is not a
Tre this is just a graph so minimum
spanning tree means we are just
converting the graph into tree means we
are trying to remove all the edges which
makes the cycle clear so here in this
example here is a cycle so definitely
what it says so this CH star is
definitely greater than equal to the CT
C might be it is equal but it is should
be always greater why because we can
have a cycle clear when we are creating
the cycle we cannot guarantee about the
cost minimum spanning tree is always
giving us the minimum value but here
whenever we include the cycle we cannot
always guarantee about the minimum cost
or minimum weight that is what we said
this is just one first expression or
mathematical model we designed or
developed which is 35.4 we named that
one clear I hope it is clearly
understood when we start with the full
work full work means we started from a
we are completing B like we are just
doing the pre-ordering which we have
said this one pre-ordering we are just
trying to go for the pre-ordering of the
full work like we started with a then b
c then B then H then again B so I must
write like how we are covering a then B
then C then again
B then H then H to again B then
a then
D then e f again e then G then D then
a clear this is what we are doing in a
full work it is also written there a b c
BH you can say a b c BH clear this is
what ba means this is exactly what I I'm
just writing this one I was just writing
this one it is written in the book this
is what a full work we are doing the
pre-ordering of it
clear like we have done just for what
reason just for find the triangle
inequality for finding triangle
inequality for triangle
inequality that's why we are just
covering all the one whichever is
greatest for us we trying to remove that
one clear
next so for that CW which is our full
work definitely 2 *
we are doing it two times how you can
say for A to B we have covered clear a B
to C we have covered for the first time
now then c2b we are covering again now
this is for one time this is for the
second time so you can say again for B
to H we have covered for one time again
H to B whenever we are coming this is
for the second time clear so whenever
I'm just coming towards this here it is
for the first time here it is for the
second time so indirectly we are just
covering all the paths second time means
this is what it represent CW full
work CW is 2
C to cost of t means the spanning tree
it is two times spanning tree okay the
cost of spanning tree two times we are
covering because of this region A to B
we have covered B to C is the first time
but whenever we coming back clear C2 B
it covers again the same uh AG cost or
the cost or the weight is covering again
that is what it represents
clear next if we equate and we name that
one 35.5 clear if we equate that one so
we will get this result as uh the the
full work which we did is equal to is
less less than equal to the cost of the
optimal answer which which we should get
you can equate that the previous two one
which we equate through that we have
finalized the answer as this clear next
what exactly we going to do we are now
saying whenever we are doing the uh
first visit pre-order pre-order case
what exactly Tells A B C then H then a d
this is what it represents a b c h then
d e FG this is for the pre PR order this
is pre-order that we have completed one
okay so once we have done
it so here you can check that one D
clear once we covered that one with F2 G
then G2 a there is another one like we
are going to the that one but how we
cover a b then B to C then C2 H H2 D D2
e E2 F F2 G this is what it is written
okay this is what we are getting just
deleting such notes which are giving us
maximum results using triangular
inequality or triangle inequality we did
that one the triangle inequality we
applied that one and we got certain
results definitely those results are
less than the full work why because we
are removing from the full work this is
our full work and we removed the full
work we have focused on minimizing that
one so definitely when we focusing on
minimizing that one so it is less than
here it is not we are saying it is two
times definitely we'll get either equal
to sign but that should be less so this
represents this is our optimal this is
our approximate value this is our full
work full work is less than equal to
this one clear whenever you equate uh we
name that one 37
35.7 so whenever you equate 3 5.6 with
35.7 because our Target is two way
approximation how it does like you can
say here CW here CW you just replace the
CW or the two CH there so we got the
value is this one this represents the
approximation value is 2 times greater
than the optimal answer this is what it
says clear so this
says this ISS two approximation clear
two approximation
I'm writing this it to
approximation because CH represents the
cost of approximation and C A Star is
optimal answer the optimal answer is two
times more clear so that's why what is
we what exactly we saying so uh the
answers which we are expecting here is
two times more than the optimal answer
which we are expecting that's why we are
saying it is two approximation algorithm
using triangle inequality clear so two
approximation as we have already told
you like this one C is two times more
than the C star okay that is what it is
written suppose I'm just writing
c 2 * C star this is what we are getting
this result so this is what we are
expecting this c h now to
c h
star clear this is what two *
approximation is so I hope it is clearly
understood by you how we have completed
with the approximation methodology we
have used the approximation indirectly
we said that triangle inequality how it
helps us to get a polom time though we
expecting the polom time here the prims
algorithm we have run so here others are
not exactly much time we expecting for
the prims algorithm PR algorithm we can
say the time complexity is e v log V
also we are saying b of e log V also we
are saying sometimes you can say it is
also r b of V plus e why because we are
just removing e Vortex with the ages
because we started with a Connect One
log V 2 this is what the time complexity
of Pol you know MST Pol PRS algorithm so
that's why we can say that complexity is
this one which is polom that's why
though it's polinomial one for us so we
must say the complexity of the algorithm
is polom so I'm just writing it clearly
time complexity
big of e you can say you can use mod
also not an
issue you can use mode or you cannot use
mode it it's up to you by the way
clear next we
of
uh we of V +
e then log V this also some someone can
write in this order also not an issue
but we must say this is the Pol
polinomial time it is taking but we are
taking the help of triangle inequality
this is what we did with the
approximation so I hope it is clearly
understood by you if any doubt you will
be having please comment below thank you
for having a good day
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