Math 3012 Combinatorial Optimization Worksheet

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1 Math 3012 Combinatorial Optimization Worksheet Combinatorial Optimization is the way in which combinatorial thought is applied to real world optimization problems. Optimization entails achieving the sufficient solution while making the best use of a given resource. The provided examples are the Traveling Salesman and Knapsack problems. These problems have applications in economics, shipping, and aviation. Traveling Salesman - Consider a salesman who must travel to a given set of homes and then return to his starting point. He wants to minimize the distance he has to travel while still visiting each home. How might he find the optimal tour? 1. Locations and distances can be represented as weighted graphs. A weighted graph is a graph where each edge has an associated cost or weight. Convert the following data into a weighted graph. A dash (-) indicates that the vertices are unconnected. A B C D E F G A B C D E F G

2 2. Brainstorm algorithms to solve this problem. What is the time complexity of your best algorithm? 3. Find the optimal tour for the graph from number 1. Finding the optimal tour in the Traveling Salesman problem can be O(n!) if you use an intuitive solution and simply permute every path and then take the minimal cost path to be your optimal path. Using dynamic programming, solutions can be found in O(n 2 *2 n ), which is still not polynomial time. We will now discuss a heuristic algorithm - an algorithm which often yields a short path but is not guaranteed to return the shortest path - called the Nearest Neighbor Algorithm. 4. Nearest Neighbor Algorithm- Start at index A and always choose to travel along the connected edge with minimal weight that connects to an unvisited vertex. Now mark this new vertex as visited and repeat until the last unvisited vertex is the starting vertex and end there. Use this algorithm to find a tour using the data below and staring at vertex D. Is this the optimal tour? If, not find a better tour. A B C D A B C D

3 5. Use the Nearest Neighbor algorithm to find a tour starting at vertex A for the graph detailed below. Is it the optimal tour? If, not find a better tour. A B C D A B C D

4 6. Oftentimes, finding short routes isn't as simple as the previous examples. Especially in cities like Atlanta, traffic is a huge factor in travel time, making the travel time from city A to city B sometimes be more than two times the travel time from city B to city A. Directed Graphs - Applying this logic, we can look at weighted directed graphs, where each edge only goes from one vertex to another and not the other way. In this case, we no longer look for cycles, and instead look for the shortest path from one vertex to another. a. This matrix maps the directed edges between vertices. The left column is where the edge originates, and the top row indicates the time it takes to take the edge to that node. For example, the edge from C directed towards A has a weight of 6. Draw this graph. From \ To A B C D E A B C D E

5 b. How many paths exist from A to B? What is the shortest path from A to B? c. How many paths exist from B to A? What is the shortest path from B to A? d. What's an optimal algorithm for finding the shortest path between two vertices? Knapsack / Packing Problem A knapsack problem is a common combinatorial optimization problem that involves finding the optimal combination of items given their weights and values under a certain weight constraint. Given values, v ", and weights, w ", of n items, maximize subject to the constraint with total weight W % "&' v " x "

6 % "&* w " x " W The salesman sells laptops, cameras, and earbuds. His sales-truck broke down, so he only carries in his carrier satchel his deliverables. Satchel capacity for total weight: 24 Laptop Camera Earbuds Weight Value What is the optimal combination of products he can pack in his satchel so that his bag contains the most value? 7. Use a greedy algorithm based on the ratio of value to weight. 8. Is the greedy algorithm the most optimal in this case? Explain why or why not. 9. Does the greedy algorithm always work for a knapsack problem? Prove if correct. If not correct, give a counterexample and give a possible relaxation in constraints that would make the greedy algorithm always optimal.

Design and Analysis of Algorithms CS404/504. Razvan Bunescu School of EECS.

Welcome Design and Analysis of Algorithms Razvan Bunescu School of EECS bunescu@ohio.edu 1 Course Description Course Description: This course provides an introduction to the modern study of computer algorithms.