Approximate solutions of Traveling Salesman Problem.

1. Traveling
Salesman
Problem
Maksym Voitko

2. Problem
Given a list of cities and the distances between each
pair of cities, what is _the shortest possible route_
that visits each city and returns to the origin city?.

3. Bonus
NP-hard - Non-deterministic Polynomial-time Hardness

4. Context
The importance of the TSP does not arise from an overwhelming
demand of salespeople to minimize their travel length.
Rather from a wealth of other applications such as:
• vehicle routing,
• circuit board drilling,
• VLSI design,
• robot control,
• X-ray crystallography,
• machine scheduling,
• and computational biology.

5. Solution
The traditional lines of attack for the NP-hard problems are
the following:
• Devising exact algorithms, which work reasonably fast
only for small problem sizes.
• Devising "suboptimal" or heuristic algorithms, i.e.,
algorithms that deliver approximated solutions in a
reasonable time.
• Finding special cases for the problem (“subproblems") for
which either better or exact heuristics are possible.

6. GREED is good

7. Heuristic and approximation
algorithms
Modern methods can find solutions for extremely large
problems (millions of cities) within a reasonable time, which
are with a high probability just 2–3% away from the optimal
solution.

8. Nearest neighbors
1.Make two sets of nodes, set A and set B, and put all
nodes into set B
2.Put your starting node into set A
3.Pick the node which is closest to the last node which was
placed in set A and is not in set A;
4.Put this closest neighbouring node into set A
5.Repeat step 3 until all nodes are in set A and B is empty.

10. Smallest Insertion
Put the node to the current
tour after the point,
where it results in the least
possible increase in the
tour length.

11. Results Time
Poins count Algorithm Nearest Neighbor Smallest Insertion
10 0.0002040863037 0.0002779960632
100 0.005702018738 0.01178383827
1000 0.5322930813 1.139199972

12. Results Tour Length
Poins count
Algorithm
Nearest Neighbor Smallest Insertion Optimal
10 1566.1363 1655.7461 1552.9612
100 7389.9297 4887.2190
1000 27868.7106 17265.6282

13. Further Plans
Ant colony optimization
1. Populate the beginning city with a colony of ants.
2. For each ant, send the ant on a tour of all cities where they visit each
city only once. The selection for which city to visit next is initially
effectively random. After the first run, the next city selection becomes
gradually more influenced by the amount of pheromone on a possible
trail.
3. Once all ants have completed a tour of the world, deposit pheromone
on each city path. The amount of pheromone deposited by each ant
is proportional to the length of the path traveled by the given ant.