Question

Consider the following problem. You are given a flow network with unit- capacity edges: It consists of a directed graph G = (V, E), a source s ∈ V, and a sink t ∈ V; and ce = 1 for every e ∈ E. You are also given a parameter k. The goal is to delete k edges so as to reduce the maximum s-t flow in G by as much as possible. In other words, you should find a set of edges F ⊆ E so that |F| = k and the maximum s-t flow in G′ = (V, E − F) is as small as possible subject to this. Give a polynomial-time algorithm to solve this problem.

Answer 1

To reduce the maximum flow in a unit-capacity flow network by deleting a specific number of edges, one could use a maximum flow algorithm and then remove a set number of critical edges from the minimum s-t cut until the required flow reduction is achieved.

Step-by-step explanation:

The problem presented is a flow network optimization problem, specifically focusing on unit-capacity edges, and can be classified within the field of combinatorial optimization.

To solve this problem within polynomial time, one may use a maximum flow algorithm like the Edmonds-Karp algorithm or the Ford-Fulkerson method.

These algorithms help to find the maximum flow from a source s to a sink t in a flow network. After determining the maximum flow, it is necessary to select the edges for deletion.

This can be done by identifying critical edges; these are the edges that, if removed, directly affect the maximum flow value.

To achieve the goal, one can implement the following steps:

1. Run a maximum flow algorithm on the original graph G to find the flow value.
2. Find the augmenting paths and the edges crucial to these paths. These edges form part of the minimum s-t cut.
3. Remove the minimum number of critical edges up until you reach k edges. If k is larger than the number of critical edges, any additional edges removed will not affect the flow value.

The critical step here is to identify which edges to remove after finding the maximum flow. This will typically involve analyzing the residual graph and looking for bottlenecks in the flow.

The implemented algorithm must also take into account the parameter k, which limits the number of edges that can be removed.