CS 217 Algorithms and Complexity Homework Assignment 2

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1 CS 217 Algorithms and Complexity Homework Assignment 2 Shanghai Jiaotong University, Fall 2015 Handed out on Due on You can hand in your solution either as a printed file or hand-written on paper (in this case please write nicely). You have to justify your solutions (i.e., provide proofs). You can solve the homework assignment in your group, and every group should hand in one solution. Do not copy solutions from other groups! If you are completely stuck, you may ask me for advice! 2 Network Flows, Matchings, and Paths 2.1 Flows Let G = (V, E, c) be a network with capacities. That is, (V, E) is a directed graph and c : E R + are capacities. We can generalize c to a function V V R + by setting c(u, v) = 0 whenever (u, v) E. 1

2 An s-t-flow in G is a function f : V V R such that f(u, v) = f(v, u) f(u, v) = 0 u V \ {s, t} v V f c (Skew symmetry) (Flow conservation) (Capacity constraints) The value of a flow is val(f) := v V f(s, v). Exercise 2.1. [Flow Conservation Properties] 1. Prove that u,v S f(u, v) = 0 for all sets S V. 2. Prove that val(f) = v V f(v, t). 3. Let S V be a set with s S and t S. The flow from S to V \ S is defined as f(s, V \ S) := u S,v S\V f(u, v). Prove that f(s, V \ S) = val(f). An s-t-cut is a set S V such that s S, t V \ S. The capacity of the cut S is c(s, V \ S) := u S,v V \S c(u, v). Exercise 2.2. [Flow is Smaller than Cut] Let f be a flow in G and S be a cut. Show that val(f) c(s, V \ S). Exercise 2.3. Prove that flow is transitive in the following sense: If there is a flow from s to r of value k, and a flow from r to t of value k, then there is a flow from s to t of value k. Hint. The solution is extremely short. If you are trying something that needs more than 3 lines to write, you are on the wrong path. 2.2 Matchings Let G = (V, E) be a graph. A matching in G is a set M E of edges such that every v V is incident to at most one edge of M. 2

3 Exercise 2.4. [Matchings in Regular Bipartite Graphs] Let G = (V, E) be a bipartite graph and V = V 1 V 2 be a bipartition. Suppose there are numbers d 1, d 2 such that deg G (v) = d 1 for all v V 1 and deg G (v) = d 2 for all v V 2. Show that G contains a matching of size min( V 1, V 2 ). Hint. For simplicity assume V 1 V 2. Consider the flow network defined on V {s, t}. Define a (non-integral) flow of value V 1 and then invoke the theorem that there is an integral maximum flow. Show that this integral flow corresponds to a matching. Let V = {0, 1} n. The n-dimensional Hamming cube H n is the graph (V, E) where {u, v} E if u, v differ in exactly one coordinate. Define the i th level of H n as L i := {u V u 1 = i}, i.e., those vertices u having exactly i coordinates which are L 3 L 1 L 2 L 0 The 3-dimensional Hamming cube and the four sets L 0, L 1, L 2, L 3. Exercise 2.5. [Matchings in H n ] Consider the induced bipartite subgraph H n [L i L i+1 ]. This is the graph on vertex set L i L i+1 where two edges are connected by an edge if and only if they are connected in H n. Show that for i n/2 the graph H n [L i L i+1 ] has a matching of size L i = ( n. 3

4 010 L 3 L 1 L 2 L 0 A matching of size 3 between L 1 and L Vertex Disjoint Paths Suppose we have a directed graph G = (V, E) but instead of edge capacities we have vertex capacities c : V R. Now a flow f should observe the vertex capacity constraints, i.e., the outflow from a vertex u should not exceed c(u): u V : f(u, v) c(u). v V,f(u,v)>0 Exercise 2.6. Consider networks with vertex capacities. 1. Show how to model networks with vertex capacities by networks with edge capacities. More precisely, show how to transform G = (V, E, c) with c : V R into a network G = (V, E, c ) with c : E R such that every s-t-flow f in G that respects the vertex capacities corresponds to an s-t-flow f (of same value) in G that respects edge capacities, and vice versa. 2. Draw a picture illustrating your solution. 3. Show that there is a polynomial time algorithm solving the following problem: Given a directed graph G = (V, E) and two vertices s, t V. Are there k paths p 1,..., p k, each from s to t, such that the paths are internally vertex disjoint? Here, internally vertex disjoint means that for i j the paths p i, p j share no vertices besides s and t. Exercise 2.7. Let H n be the n-dimensional Hamming cube. For i < n/2 consider L i and L n i. Note that L i = ( ) ( n i = n n = Ln i, so the L i and L n i have the same size. 4

5 Show that there are ( n paths p1, p 2,..., p ( n in H n such that ( each p i starts in L i and ends in L n i ; (i two different paths p i, p j do not share any vertices. Hint. Model this problem as a network flow with vertex capacities. Then define a (non-integral) flow of value ( n. Invoke the fact that there exists an integral maximum flow and show how this flow corresponds to a collection of ( n paths from Li to L n i. Notes. Exercise 2.5 in particular shows the following: Suppose n is odd and let i = n 1 n+1. Then i + 1 = n i =. Since L 2 2 i = L i+1 the exercise tells us that there is a perfect matching between these two sets. However, constructing such a matching explicitly (whatever this precisely means) is a non-trivial task (there are papers about it). Also, note that for n = 3 the graph H 3 [L 1 L 2 ] has not only a perfect matching but also a Hamilton cycle. In fact, the question whether H 2k+1 [L k L k+1 ] has a Hamilton cycle for every k is known as the Midlevel Conjecture. It has been open for 30 years and was proved by Torsten Mütze in