The document discusses various combinatorial optimization problems including the minimum spanning tree (MST), travelling salesman problem (TSP), and knapsack problem. It provides details on the MST and TSP, defining them, describing algorithms to solve them such as Kruskal's and Prim's for the MST and dynamic programming for the TSP, and discussing their applications and time complexities. The document also compares Prim and Kruskal algorithms and discusses how dynamic programming can provide an efficient solution for the TSP in some cases but not when the number of targets is too large.

1. COMBINATORIAL
OPTIMIZATION
Guided By: Prepared By
Dhaval Jha Anjali Bidua
Asst Professor 12bce155

2. Overview
What is Optimization?
What is Combinatorial Optimization?
Applications of Combinatorial Optimization
Identification of different Combinatorial Problems
Minimum Spanning Tree
Overview Of Travelling Salesman Problem

3. Optimization
Optimization is technique of finding the optimal solution when
more than one solutions are available.
It is a live field of Mathematics.
It can be better known by Integer Programming.
Linear programming is a mathematical programming technique
to optimize performance under a set of resource constraints.

4. There are various methods for solving optimization problems by
integer programming.
Some of them are:
Graphical Method
Simplex Method

5. Combinatorial Optimization
Combinatorial optimization deals with algorithmic
approaches to finding specified configurations or
objects in finite structures such as directed and
undirected graphs, hyper graphs, networks, matroids,
partially ordered sets, and so forth.

6. Major Combinatorial Optimization Problems
Minimum spanning tree
Travelling salesman problem
Vehicle routing problem
Weapon target assignment problem
Knapsack problem

7. Minimum Spanning Tree
Given a connected, undirected graph, a spanning tree
of that graph is a subgraph that is a tree and connects
all the vertices together.
A minimum spanning tree (MST) or minimum weight
spanning tree is then a spanning tree with weight less
than or equal to the weight of every other spanning
tree.

8. Travelling Salesman Problem
Given a set of cities and the distance between each
possible pair, the Travelling Salesman Problem is to
find the best possible way of ‘visiting all the cities
exactly once and returning to the starting point’.

9. Vehicle Routing Problem
The vehicle routing problem (VRP) is a combinatorial optimization and
integer programming problem seeking to service a number of customers
with a fleet of vehicles.
VRP is an important problem in the fields of transportation, distribution and
logistics. Often the context is that of delivering goods located at a central
depot to customers who have placed orders for such goods.
Implicit is the goal of minimizing the cost of distributing the goods. Many
methods have been developed for searching for good solutions to the
problem, but for all but the smallest problems, finding global minimum for
the cost function is computationally complex.

10. Weapon Target Assignment problem
It consists of finding an optimal assignment of a set of
weapons of various types to a set of targets in order
to maximize the total expected damage done to the
opponent

11. Knapsack problem
The problem often arises in resource allocation
where there are financial constraints and is
studied in fields such as combinatorics,
computer science, complexity theory,
cryptography and applied mathematics.

12. Minimum Spanning Tree
Definition: A tree is a connected undirected graph with no
simple circuits.
A spanning tree of a graph is just a subgraph that contains all
the vertices and is a tree.
Spanning Tree properties: On a connected graph G=(V, E), a
spanning tree:
• is a connected subgraph
• acyclic
• is a tree (|E| = |V| - 1)
• contains all vertices of G (spanning)

13. Minimum weight spanning tree (MST)
On a weighted graph, A MST:
connects all vertices through edges with least weights.
has w(T) ≤ w(T’) for every other spanning tree T’ in G
Adding one edge not in MST will create a cycle

14. Finding Minimum Spanning Tree
Algorithms :
Kruskal’s
Prim’s

15. Compare Prim and Kruskal
Complexity
Kruskal: O(NlogN)
comparison sort for edges
Prim: O(NlogN)
search the least weight edge for
every vertice

16. MST Conclusion
Kruskal’s has better running times if the number
of edges is low, while Prim’s has a better running
time if both the number of edges and the
number of nodes are low.
So, of course, the best algorithm depends on the
graph and if you want to bear the cost of
complex data structures.

17. Applications of MST
Design of networks, including computer networks,
telecommunications networks, transportation
networks, water supply networks, and electrical grids
Applications of MST in Sensor Networks
A tree is formed from all sensor nodes to the sink so
that the transmission power of all sensor nodes is
minimum

18. Travelling Salesman Problem
In the theory of computational complexity, the
decision version of the TSP where, given a length L,
the task is to decide whether the graph has any tour
shorter than L.

19. TSP : An NP-HARD Problem!
TSP is an NP-hard problem in combinatorial optimization
studied in theoretical computer science.
In many applications, additional constraints such as limited
resources or time windows make the problem
considerably harder.
Removing the constraint of visiting each city exactly one
time also doesn't reduce complexity of the problem.

20. Applications of TSP
Planning
Logistics
Manufacture of microchips
Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing
Integrated circuit
Mechanical arm

21. Solutions
Exact solutions
Heuristic algorithms
Finding subproblems for the problem

22. Exact Solution
Consider city 1 as the starting and ending point.
Generate all (n-1)! Permutations of cities.
Calculate cost of every permutation and keep track of
minimum cost permutation.
Return the permutation with minimum cost.
Time Complexity: O(n!)

23. Heuristic Solution
The greedy algorithm can be used to give a good
but not optimal solution in a reasonably short
amount of time.
The greedy algorithm heuristic says to pick
whatever is currently the best next step
regardless of whether that precludes good steps
later.

24. Dynamic Programming
Division of Problem into Sub-Problems
Memoization

25. An Example

26. An Example
Let the given set of vertices be {1, 2, 3, 4,….n}.
Let us consider 1 as starting and ending point of
output.
For every other vertex i (other than 1), we find
the minimum cost path with 1 as the starting
point, i as the ending point and all vertices
appearing exactly once.

27. Let the cost of this path be cost(i), the cost of corresponding
Cycle would be cost(i) + dist(i, 1) where dist(i, 1) is the distance
from i to 1. Finally, we return the minimum of all [cost(i) + dist(i,
1)] values.
Let us define a term C(S, i) be the cost of the minimum cost
path visiting each vertex in set S exactly once, starting at 1 and
ending at i.
We start with all subsets of size 2 and calculate C(S, i) for all
subsets where S is the subset, then we calculate C(S, i) for all
subsets S of size 3 and so on. Note that 1 must be present in
every subset.

28. If size of S is 2, then
S must be {1, i}, C(S, i) = dist(1, i)
Else if size of S is greater than 2
C(S, i) = min { C(S-{i}, j) + dis(j, i)}
where j belongs to S, j != i and j != 1.
For a set of size n, we consider n-2 subsets each of size n-1 such that all
subsets don’t have nth in them.
The total running time is O(n2*2n). The time complexity is much less
than O(n!), but still exponential. Space required is also exponential. So
this approach is also infeasible even for slightly higher number of
vertices.

29. TSP Using Dynamic Programming

31. TSP Without Dynamic
Programming

33. Sparse Graphs
For 2 cities

34. With Dynamic Programming

35. Without Dynamic Programming

36. Dense Graphs
For 6 cities

37. With Dynamic Programming

38. Without dynamic Programming

39. Conclusion
• When we have a limited set of targets to
meet, Tsp using Dynamic Programming is
efficient.
• But when we have a considerable more
number of targets this concept is not much
useful.

40. Various scenarios

41. Connected Graph

42. Complete Graph

43. References
• Discrete Optimization". Elsevier. Jump up Jump up Cook,
William. "Optimal TSP Tours". University of Waterloo.
• Fredman, M. L.; Willard, D. E. (1994), "Trans-dichotomous
algorithms for minimum spanning trees and shortest
paths", Journal of Computer and System Sciences
• Karger, David R.; Klein, Philip N.; Tarjan, Robert E. (1995), "A
randomized linear-time algorithm to find minimum
spanning trees", Journal of the Association for Computing
Machinery
• Applegate, D. L.; Bixby, R. M.; Chvátal, V.; Cook, W. J. (2006),
The Traveling Salesman Problem
• Bellman, R. (1962), "Dynamic Programming Treatment of
the Travelling Salesman Problem", J. Assoc. Comput. Mach.