Let's answer the questions step by step as requested: --- ### Step 1: **(a) Explain how the flow value evolves over the course of this sequence. Does the flow value converge to the max flow?** #### **Explanation:** - **Initial Path (A):** The first path used is \( A: s \to a \to b \to t \). The bottleneck edge along this path is the edge with capacity \( \varphi \) (the edge \( a \to b \)), so \( \varphi \) units of flow are sent along this path. - **Subsequent Paths (BCBD):** The next paths repeatedly used are \( BCBD: s \to b \to c \to b \to d \to t \) (looping through \( b \) and \( c \)). Each time, the amount of flow that can be pushed along each \( BCBD \) path is a fraction of what was pushed before, due to the non-integer residual capacities left after each augmentation. Specifically, the residual capacity decreases by a factor of \( \varphi \) each time because of how the cycles and capacities interact. This is similar to a geometric series: - After the first \( BCBD \), the flow remaining on the critical edges is multiplied by \( \varphi \). - After \( k \) augmentations, the total added flow is \( \varphi + \varphi^2 + \varphi^3 + \ldots + \varphi^k \). - **Does it converge?** - Yes, the total flow converges to a limit, since it is a geometric series with ratio \( \varphi < 1 \). - **However, it never actually reaches the true max flow in a finite number of steps**, because the sum \( \varphi + \varphi^2 + \varphi^3 + \ldots \) approaches a limit but never attains it exactly. #### **Conclusion after Step 1:** The flow value increases by a geometric series and converges to a value less than the true max flow, so **the Ford-Fulkerson algorithm may not terminate on this graph with non-integer capacities**. --- ### Step 2: **(b) Could we have picked a different set of paths to use so that we indeed find the max flow? Write a rule for choosing which path Ford-Fulkerson should use in every iteration so that it does find the correct max flow for this particular graph.** #### **Explanation:** - The issue with the sequence \( A, BCBD, BCBD, ... \) is that it always chooses paths that push smaller and smaller amounts of flow due to the non-integer capacities and cycles, never saturating the edges that would help us reach max flow. - **If, instead, we always pick the augmenting path with the largest available residual capacity (i.e., the "fattest" path or the path that can carry the most flow from \( s \) to \( t \) in the current residual graph),** we can guarantee that the number of augmentations is finite and the algorithm will terminate with the correct max flow. - This approach is called the **Edmonds-Karp algorithm**, which always selects the shortest augmenting path (in terms of number of edges), or more generally, always picks the path that maximizes the bottleneck capacity in each iteration. #### **Rule for Path Selection:** - **At each step, choose the augmenting path from \( s \) to \( t \) in the residual graph with the largest possible bottleneck (minimum) capacity.** - Alternatively, always pick the shortest augmenting path if all edge capacities are positive. #### **Conclusion after Step 2:** **Yes, by always choosing the path with the largest available capacity (or the shortest path), Ford-Fulkerson will terminate with the correct max flow.** --- ## **Final Answer** --- ### **(a)** The flow value increases as a geometric series because each augmenting path (after the first) adds a smaller fraction of flow due to the non-integer capacities. The total flow approaches a limit but never reaches the true max flow in a finite number of steps. **Thus, the value converges, but Ford-Fulkerson does not terminate with the max flow for this sequence of paths.** ### **(b)** Yes, if we always pick the augmenting path with the largest available capacity (the path with the largest bottleneck), **Ford-Fulkerson will terminate with the correct max flow.** The rule is: **At each iteration, choose the augmenting path from \( s \) to \( t \) in the residual graph with the largest possible bottleneck capacity.**