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![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.](https://image.slidesharecdn.com/combinatorialoptimizationfinal-140920135113-phpapp02/85/Combinatorial-Optimization-27-320.jpg)
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Combinatorial Optimization
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.
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Combinatorial Optimization
- 1. COMBINATORIAL OPTIMIZATION GuidedBy: Prepared By Dhaval Jha Anjali Bidua Asst Professor 12bce155
- 2. Overview What isOptimization? What is Combinatorial Optimization? Applications of Combinatorial Optimization Identification of different Combinatorial Problems Minimum Spanning Tree Overview Of Travelling Salesman Problem
- 3. Optimization Optimization istechnique 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 variousmethods for solving optimization problems by integer programming. Some of them are: Graphical Method Simplex Method
- 5. Combinatorial Optimization Combinatorialoptimization 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 OptimizationProblems 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 Assignmentproblem 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 Theproblem 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 spanningtree (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 SpanningTree Algorithms : Kruskal’s Prim’s
- 15. Compare Prim andKruskal Complexity Kruskal: O(NlogN) comparison sort for edges Prim: O(NlogN) search the least weight edge for every vertice
- 16. MST Conclusion Kruskal’shas 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 : AnNP-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 Considercity 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 Thegreedy 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 Divisionof Problem into Sub-Problems Memoization
- 25.
- 26. An Example Letthe 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 costof 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 ofS 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 DynamicProgramming
- 31. TSP Without Dynamic Programming
- 33. Sparse Graphs For2 cities
- 34.
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- 36. Dense Graphs For6 cities
- 37.
- 38.
- 39. Conclusion • Whenwe 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.
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- 43. References • DiscreteOptimization". 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.