Network Flow. By: Daniel Pham, Anthony Huber, Shaazan Wirani

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1 Network Flow By: Daniel Pham, Anthony Huber, Shaazan Wirani

2 Questions 1. What is a network flow? 2. What is the Big O notation of the Ford-Fulkerson method? 3. How to test if the Hungarian algorithm is optimal?

3 Anthony Huber Computer Science senior Hololens research under Dr. Huang Hometown: Elizabethton, TN Interests: Guitar Video games with friends and family Swimming

4 Daniel Pham Computer Science senior Interests: Gaming Working out Hometown: Nashville, TN

5 Shaazan Wirani Major: Computer Science Chattanooga, TN Minor: Mathematics Originally from India Interests: Sports: Football and Basketball Spending time with my family

6 Outline History of Flow Network Overview of Flow Network Definitions of the Algorithms with Applications Implementations Open Issues References Discussion

7 1986: Andrew Goldberg and Robert Tarjan History Ford-Fulkerson Method: First published in 1956 by: Lester Randolph Ford Jr. Delbert Ray Fulkerson Assignment Problem: 1955: Harold Kuhn Matrix implementation 1987: Harold Gabow and Robert Tarjan Minimum Cost Circulation: 1972: Jack Edmonds and Richard Karp 1987: Andrew Goldberg, Robert Tarjan and Orlin 1988: Ravindra Ahuja, Andrew Goldberg, Orlin, and Robert Tarjan Generalized Flow: 1989: Vaidya Maximum flow:

8 Network Flow What is Network Flow: A network flow is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge.

9 A B D E : Sending 2 Mb/s We get two new Graphs

10 Residual Graph and Augmenting Graph

11 Ford-Fulkerson for Maximum Flow Greedy algorithm that starts from an empty flow and as long is it can find an augmenting path, updates the current solution Each augmenting path can be found in O(E) time To find the maximum flow, increasing the flow by an integer 1 for each flow, but is bounded by the total number of flows which is f O(f E)

12 Edmonds-Karp Algorithm Variation of the Ford-Fulkerson method The difference between these two methods are that augmenting path is defined. O(f E) Ford-Fulkerson Method O(V E 2 ) Edmonds - Karp Method The augmenting path is the shortest path that has available capacity (the shortest path in the residual graph) This path can be found with a BFS

13 Maximum Bipartite matching A Bipartite Graph is a graph whose vertices can be divided into two disjoint sets U and V such that each edge connects a vertex U in one of V Example :

14 The Assignment Problem A general problem layout: The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agenttask assignment. It is required to perform all tasks by assigning exactly one agent to each task and exactly one task to each agent in such a way that the total cost of the assignment is minimized.

15 The Hungarian Algorithm The algorithm applies to a given NxN cost matrix to find the optimal assignment. 1. Subtract the smallest entry in each row from all the entries in the row 2. Subtract the smallest entry in each column from all the entries in the column 3. Draw lines through the appropriate rows using the least number of horizontal and vertical lines to covering the 0 s of the matrix 4. Test for optimality : if the minimum number of lines is N -->found

16 Example: Matrix Implementation Example 2: A construction company has four large bulldozers located at four different garages. The bulldozers are to be moved to four different construction sites. How should the bulldozers be moved to the construction sites in order to minimize the total distance traveled?

17 Step 1: Subtract the smallest entry in each row from all the entries in the row

18 Step 2: Subtract the smallest entry in each column from all the entries in the column

19 Step 3: Draw lines through the appropriate rows using the least number of horizontal and vertical lines to covering the 0 s of the matrix

20 Step 4: Test for optimality If the number of vertical and horizontal lines equals N then the optimized matrix is found. Since the minimal number of lines is less than 4, we have to proceed to Step 5.

21 Step 5: Determine the smallest entry not covered by any line and subtract it from each uncovered row and add it to each covered column. Then return to step 3

22 After 2 more iterations 1: Diagonal: = 275 D1 + C2 + B3 + A4 2: Another way: = 275 B1 + D2 + C3 + A4 3: NonZero: = 380 A1 + B2 + C3 + D4

23 Algorithms/Methods Ford-Fulkerson Edmonds-Karp Maximum Bipartite Matching Assignment / Hungarian Algorithm Generalized Network Flow

24 More of a method than an algorithm Ford-Fulkerson Find the maximum flow through a graph. The following is the simple idea of the Ford-Fulkerson algorithm: 1) Start with initial flow as 0. 2) While there is an augmenting path from source to sink. Add this path-flow to flow. 3) Return flow. Edmonds-Karp algorithm defines a way to find the augmenting path. Use BFS finding shortest path with available capacity.

25 Edmonds-Karp source sink

26 Maximum Bipartite Matching With a bipartite graph on input, find the maximum matching.

27 Add a source and sink. Connect the source with all applicants, and connect all jobs with the sink. Every edge is marked with a capacity of 1. Using Edmonds-Karp algorithm, find the maximum flow.

28 Assignment The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment. It is required to perform all tasks by assigning exactly one agent to each task and exactly one task to each agent in such a way that the total cost of the assignment is minimized.

Hungarian Algorithm Step 0) A. For each vertex from left part (workers), find the minimal outgoing edge and subtract its weight from all weights connected with this vertex.

29 Hungarian Algorithm Step 0) A. For each vertex from left part (workers), find the minimal outgoing edge and subtract its weight from all weights connected with this vertex. This will introduce 0-weight edges (at least one). B. Apply the same procedure for the vertices in the right part (jobs). Actually, this step is not necessary, but it decreases the number of main cycle iterations.

30 Step 1) A. Find the maximum matching using only 0-weight edges (for this purpose you can use max-flow algorithm, augmenting path algorithm, etc.). B. If it is perfect, then the problem is solved. Otherwise find the minimum vertex cover V (for the subgraph with 0-weight edges only), the best way to do this is to use Köning s graph theorem (According to this resource).

31 Step 2) Adjust the weights using the following rule where V is the set of vertices in the vertex cover: Step 3) Repeat Step 1 until solved. Time complexity: O(n^4)

32 Generalized Network Flow Generalized network flow models are achieved when the edges have an associated gain or loss. Most real world networks will have this behavior. For example: voltage transportation, water canal evaporation, currency exchange, shipping losses, etc.

33 Implementations A Ford-Fulkerson algorithm(red) Edmonds Karp algorithm(black) MIT s algorithm (Purple) - At 750 nodes, over 8 times better - than the other two

35 Implementations: Ford-Fulkerson/Edmonds Karp Greedy algorithm checks all paths to find an augmenting path, updates the total flow Checks a path, then tries to find a path that maximizes the flow Takes more time as the size of the graph increases Recent years, the size of graphs being studied increased

36 Implementations: MIT s Algorithm Finding an efficient way to move through a network (i.e. transportation of materials, internet traffic) becomes increasingly problematic as the size of the network increases Reduces the number of operations Take less unnecessary actions

37 Implementations: MIT s Algorithm Finds areas of the graph that create a bottleneck effect MIT professors and graduate students viewed the graph like an electric resistor Divides the graph into clusters of well connected nodes The paths between these clusters, are the paths that create a bottleneck Focus on higher level problems/structures, less on unimportant decisions MIT s algorithm is based off of max flow, with almost linear time operation According to an MIT representative, the amount of time it would take to solve a problem is almost directly proportional to number of nodes in the network

38 Open Issues Multi-commodity flow problem Multiple sinks/sources Real world applications of the algorithms Correctness of different algorithms(i.e. returning incorrect max flow, preferences) Optimization The time to compute

39 Open Issues: multi-commodity flow Multiple sources/sinks How do we know which source to choose first? Algorithms are already time consuming on large graphs with one source/sink Even more costly on time and computation to check all paths in a large graph with multiple sources/sinks How can this be addressed?

40 Open Issues: multi-commodity flow Supersource - vertex connecting all sources Supersink - vertex connecting all sinks Not a realistic approach Infinite amount of flow from one point

Open Issues: Real World Applications Algorithms don t take the constraints of the real world How do we factor cost/efficiency while maximizing flow?

41 Open Issues: Real World Applications Algorithms don t take the constraints of the real world How do we factor cost/efficiency while maximizing flow? Physical design of the path Long path with higher capacity vs short path lower capacity Long path length 10 with 2 capacity or short path length 1 and 1 capacity

42 References Fulkerson_algorithm

43 Discussions Any Questions?

44 Questions 1. What is a network flow? 2. What is the Big O notation of the Ford-Fulkerson method? 3. How to test if the Hungarian algorithm is optimal?

45 Thank You

Maximum flows & Maximum Matchings

Chapter 9 Maximum flows & Maximum Matchings This chapter analyzes flows and matchings. We will define flows and maximum flows and present an algorithm that solves the maximum flow problem. Then matchings