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Unformatted text preview: CS 473: Algorithms, Fall 2010 HBS 8
Problem 1. [Go With the Flow] The ﬁgure below shows a ﬂow network along with a ﬂow. In the ﬁgure, the notation a/b for an edge means that the ﬂow on the edge is a and the capacity of the edge is b.
4/4 a b 9/13 6/10 3/4 s c 5/7 2/2 d 7/7 e 2/4 5/8 0/3 t 2/2 5/5 (a) What is the value of the given ﬂow? Is it maximal? Show the residual graph for the above graph and ﬂow in the ﬁgure below. a b s c t d e (b) Show an s − t path in the residual graph and state its bottleneck capacity. You only need to draw the path from the graph you showed in (a). a b s c t d e 1 (c) Show the new ﬂow on the original graph after augmenting on the path you found in (b). Use the notation a/b to indicate the ﬂow on an edge and its capacity.
4/4 a b 9/13 6/10 3/4 s c 5/7 2/2 d 7/7 e d e 2/4 5/8 0/3 t s c t a b 2/2 5/5 (d) What is the capacity of a minimum-cut in the given graph? Find a cut with that capacity. Problem 2. [Residual Graph Properties] Prove the following property about residual graphs: Let f be a ﬂow in G and Gf be the residual graph. If f is a ﬂow in Gf , then f + f is a ﬂow in G of value v (f ) + v (f ). Problem 3. [Capacities on Nodes] In a standard s − t maximum ﬂow problem, we assume that edges have capacities, and there is no limit on how much ﬂow is allowed to pass through a node. In this problem, we consider the variant where nodes have capacities. Let G = (V, E ) be a directed graph with source s and sink t. Let c : V → R+ be a capacity function. Recall that a ﬂow f assigns a ﬂow value f (e) to each edge e. A ﬂow f is feasible if the total ﬂow into every vertex v is at most c(v ): f in (v ) ≤ c(v ) for every vertex v Design a polynomial time algorithm that ﬁnds a feasible s − t ﬂow of maximum value in G. 2 ...
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This note was uploaded on 01/22/2011 for the course CS 473 taught by Professor Chekuri,c during the Fall '08 term at University of Illinois, Urbana Champaign.
- Fall '08