Network Flow Problems: Ford-Fulkerson Algorithm and Min-Cost Max-Flow, Study notes of Computer Science

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Network Flow Problems

Jaehyun Park

CS 97SI

Stanford University

June 29, 2015

Outline

Network Flow Problems

Ford-Fulkerson Algorithm

Bipartite Matching

Min-cost Max-flow Algorithm

Network Flow Problems 2

Network Flow Problem

◮ Settings: Given a directed graph G = ( V, E ), where each edge

e is associated with its capacity c ( e ) > 0. Two special nodes

source s and sink t are given ( s 6 = t )

◮ Problem: Maximize the total amount of flow from s to t

subject to two constraints

• Flow on edge e doesn’t exceed c ( e )
• For every node v 6 = s, t , incoming flow is equal to outgoing flow

Network Flow Problems 4

Network Flow Example (from CLRS)

◮ Capacities

◮ Maximum flow (of 23 total units)

Network Flow Problems 5

Minimum Cut Example

◮ Capacities

◮ Minimum Cut (red edges are removed)

Network Flow Problems 7

Flow Decomposition

◮ Any valid flow can be decomposed into flow paths and

circulations

• s → a → b → t : 11
• s → c → a → b → t : 1
• s → c → d → b → t : 7
• s → c → d → t : 4

Network Flow Problems 8

Ford-Fulkerson Algorithm

◮ A simple and practical max-flow algorithm

◮ Main idea: find valid flow paths until there is none left, and

add them up

◮ How do we know if this gives a maximum flow?

• Proof sketch: Suppose not. Take a maximum flow f ⋆^ and “subtract” our flow f. It is a valid flow of positive total flow. By the flow decomposition, it can be decomposed into flow paths and circulations. These flow paths must have been found by Ford-Fulkerson. Contradiction.

Ford-Fulkerson Algorithm 10

Back Edges

◮ We don’t need to maintain the amount of flow on each edge

but work with capacity values directly

◮ If f amount of flow goes through u → v , then:

• Decrease c ( u → v ) by f
• Increase c ( v → u ) by f

◮ Why do we need to do this?

• Sending flow to both directions is equivalent to canceling flow

Ford-Fulkerson Algorithm 11

Analysis

◮ Assumption: capacities are integer-valued

◮ Finding a flow path takes Θ( n + m ) time

◮ We send at least 1 unit of flow through the path

◮ If the max-flow is f ⋆ , the time complexity is O (( n + m ) f ⋆ )

• “Bad” in that it depends on the output of the algorithm
• Nonetheless, easy to code and works well in practice

Ford-Fulkerson Algorithm 13

Computing Min-Cut

◮ We know that max-flow is equal to min-cut

◮ And we now know how to find the max-flow

◮ Question: how do we find the min-cut?

◮ Answer: use the residual graph

Ford-Fulkerson Algorithm 14

Computing Min-Cut

◮ Mark all nodes reachable from s

• Call the set of reachable nodes A

◮ Now separate these nodes from the others

• Cut edges going from A to V − A

Ford-Fulkerson Algorithm 16

Computing Min-Cut

◮ Look at the original graph and find the cut:

◮ Why isn’t b → c cut?

Ford-Fulkerson Algorithm 17

Bipartite Matching

◮ Settings:

• n students and d dorms
• Each student wants to live in one of the dorms of his choice
• Each dorm can accommodate at most one student (?!) ◮ (^) Fine, we will fix this later...

◮ Problem: find an assignment that maximizes the number of

students who get a housing

Bipartite Matching 19

Flow Network Construction

◮ Add source and sink

◮ Make edges between students and dorms

• All the edge weights are 1

Bipartite Matching 20