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DMOR Networks

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Graphs: Koenigsberg bridges Leonard Euler problem (1736)

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Nodes

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Edges

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Is there a walk which uses each edge exactly once?

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Build two more bridges.

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Each such walk has to enter and leave any given node. The degree of any node has to be even.

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Eulerian cycle Hamiltonian cycle Travelling salesman problem

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TSP - cycle Duration: 2 d 7 h 2 m Length: 3615 km

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TSP - trail Duration: 2 d 4 h 37 m Length: 3436 km

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TSP – a real walk Duration: 29 d 11 h 21 m Length: 3485 km

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Travelling salesman problem TSP Mathematically Variable x ij = 1, if the salesman passes from i to j, 0 otherwise Going from one city to the same city is forbidden Is this all ???

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TSP Is this all? Let’s say our solution with 5 cities is x 12 =x 23 =x 31 =x 45 =x 54 =1 It satisfies all the constraints. But it involves subtours (cycles involving fewer than all cities) We need to introduce additional constraints

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Subtour elimination For two cities For three cities For four cities Etc. In practical implementation too many constraints: with 30 cities there would be 870 constraints eliminating only subtours of length 2

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Subtour eleimination – second approach Introduce nonnegative variables u i : Subtours are eliminated How many such constraints? (N-1) 2 -N, i.e. with 30 cities there would be 812 constraints.

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Game http://www.tsp.gatech.edu/games/index.html

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Introduction to networks Two main elements: – arcs/edges – nodes A graph is a structure consisting of nodes and arcs bewteen nodes A directed graph (a digraph) is a graph in which arcs have a given direction A network is a graph (or digraph), in which arcs have a flow assigned to it Simple examples: NodesArcsFlow CitiesHighwaysCars HubsWire Transmitted packets Pipelines jointsPipesWater

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Introduction to networks Chain is a sequence of arcs connecting two nodes i and j, e.g. figure on the right: ABCE, ADCE Path is a sequence of directed arcs connecting two nodes, e.g. figure on the right ABDE, but not ABCE Cycle is a chain, which connects a node with the same node without any repetition (retracing) e.g. figure on the right ABCEDA, but not ABCDECBA Connected graph/network has only one part GraphDirected graph

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Introduction to networks Tree – a connected graph which does not have cycles. Spanning tree is a tree chosen among arcs in the graph so that all nodes in the tree were connected Two spanning trees Two trees Flow capacity – upper (sometimes also lower) limit for the flow at a given arc in the network, e.g. maximal number of cars that can pass per minute at a given road Source is a node which introduces a flow into the network Sink is a node which takes the flow out of the network

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The shortest route problem Formulation: For a given graph in which every arc is assigned with a distance bewteen the two nodes it connects, what is the shortest path between node i and j. Example: What is the shortest route bewteen A and H? Enumeration – impractical Dijkstra algorithm

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http://optlab- server.sce.carleton.ca/PO Animations2007/Dijkstras Algo.html

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Minimum spanning tree Formulation: For a given graph, in which every arc is assigned with a distance between the two nodes it connects, find a spanning tree that has minimal total length. Example: Find minimal length for a wire that connects all the offices in the building when all the available wire paths are given. Algorithm: An example of the so called greedy algorithm – it does what’s best in a given step not looking at the other stages of the problem (usually ineffective – here it is effective!) One can do a maximal spanning tree the same way http://optlab- server.sce.carleton.ca/POAnimations 2007/MinSpanTree.html

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Maximum flow and the minimum cut Formulation: What is the maximal flow between two given nodes in a graph? Each node is assigned with a maximal flow. Example: Find a maximal flow of cars from an underground parking lot downtown to the motorway entrance. Each arc is assigned with a maximal simultaneous flow between the two nodes it connects – Maximal flow may differ depending on the flow direction (e.g. one-way streets) Example solution: 4 cars/m on route A-D-E-G 3 cars/m ojn route A-B-E-G 4 cars/m on route A-C-F-G Total flow between A and G is 11 cars/m Is this optimal???

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Maximum flow and the minimum cut Algorithm: Ford and Fulkerson (Canadian Journal of Mathematics 1956) http://optlab- server.sce.carleton.ca/POAnimations2007/MaxFlow.html

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In Ford Fulkerson algorithm, why do we need to add flow in the opposite direction? – Accounting convention that keeps track of the flow that, if necessary, can be reversed. Maximum flow/minimum cut

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Maximum flow is closely related to minimum cut: – A cut in a graph is a set of directed arcs which contains at least one arc in every possible path from the source to the sink. If we remove arcs from a given cut, the flow from the source to the sink will no longer be possible. – Cut value is the sum of all flow capacities (direction from the source to the sink) for each arc in a cut. Possible cuts with cut values indicated on them Maximum flow/minimum cut

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