Flow Reductions Flashcards
What is the importance of choosing good augmenting paths in the Ford-Fulkerson algorithm?
“Choosing good augmenting paths improves the efficiency of the algorithm by minimizing the number of iterations required to reach the maximum flow.”
What is the bottleneck capacity of an augmenting path in the Ford-Fulkerson algorithm?
“It is the smallest residual capacity of the edges along the path.”
Explain why bad choices of augmenting paths can lead to inefficiency in the Ford-Fulkerson algorithm.
“Bad choices can cause the algorithm to make only minimal progress in each iteration
What is the Scaling Max-Flow algorithm?
“An implementation of the Ford-Fulkerson algorithm that uses scaling parameters to focus on paths with larger bottleneck capacities
What is the advantage of using scaling in the Max-Flow problem?
“It reduces the number of augmentations needed by ensuring that large bottleneck paths are prioritized in each scaling phase.”
How does the Preflow-Push algorithm differ from the Ford-Fulkerson algorithm?
“Preflow-Push operates on a preflow
What are the two main operations in the Preflow-Push algorithm?
“Push operations
What is a compatible labeling in the context of the Preflow-Push algorithm?
“A labeling that ensures no steep edge in the residual graph has its head lower than its tail by more than 1 unit.”
Describe the relationship between augmenting paths and alternating paths in bipartite matching.
“Augmenting paths in the flow network correspond to alternating paths in the bipartite graph
What is Hall’s theorem in the context of bipartite matching?
“A bipartite graph has a perfect matching if and only if every subset of nodes on one side is matched to an equal or larger subset on the other side.”
Design a bipartite matching problem where Hall’s theorem fails. Why does it fail?
“Example: X={x1
Explain the use of Max-Flow in solving the Bipartite Matching Problem.
“Max-Flow assigns unit capacity edges from a source to nodes in one partition
How is image segmentation modeled as a Max-Flow problem?
“Pixels are connected in a graph
What does the capacity of the cut represent in image segmentation?
“The capacity represents the cost of separating the image into foreground and background regions while minimizing smoothness penalties.”
Why is the Project Selection Problem reducible to a minimum cut problem?
“Precedence constraints can be modeled as infinite-capacity edges
