Let ƒ be an s,t-flow in an s, t-network D = (V, A) with capacities c : A → R>0; and assume that there is no augmenting path. Let X be the set of vertices that can be reached from s by unsaturated paths. Let (a,b) E A(X,V\ X). Explain why f(a, b) = c(a, b).
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- Consider a directed graph G = (V, E), and two distinct vertices u, v V. Recall that a set of U-V paths is non-overlapping if they have no edges in common among them, and a set C of edges disconnects from U if in the graph (V, E-C) there is no path from U to V. Suppose we want to show that for any set of non-overlapping paths P and any disconnecting set C, |P| ≤ |C|. Consider the proof that defines A = P, B = C and f(path q) = qC, and applies the Pigeonhole Principle to obtain the result. True or False: f is a well-defined function (i.e. it satisfies the 3 properties of a well- defined function). True FalseTrue or False Let G be an arbitrary flow network, with a source s, a sink t, and a positiveinteger capacity ceon every edge e. If f is a maximum s −t flow in G, then f saturates every edge out of s with flow (i.e., for all edges e out of s, we have f (e) = ce).For a directed graph G = (V,E) (source and sink in V denoted by s and t respec- tively) with capacities c: E→+, and a flow f: E→, the support of the flow f on G is the set of edges E:= {e E| f(e) > 0}, i.e. the edges on which the flow function is positive. Show that for any directed graph G = (V,E) with non-negative capacities e: Ethere always exists a maximum flow f*: E→+ whose support has no directed cycle.
- 4. Let G (V, E) be a directed graph. Suppose we have performed a DFS traversal of G, and for each vertex v, we know its pre and post numbers. Show the following: (a) If for a pair of vertices u, v € V, pre(u) < pre(v) < post(v) < post(u), then there is a directed path from u to v in G. (b) If for a pair of vertices u, v € V, pre(u) < post(u) < pre(v) < post(v), then there is no directed path from u to v in G.Consider the flow network G shown in figure 1 with source s and sink t. The edge capacities are the numbers given near each edge. (a) Find a maximum flow in this network. Once you have done this, draw a copy of the original network G and clearly indicate the flow on each edge of G in your maximum flow. (b) Find a minimum s-t cut in the network, i.e. name the two (non-empty) sets of vertices that define a minimum cut. Also, say what its capacity is. 3 2 b 6 Fig. 1 A flow network, with sources and sink t. The numbers next to the edges are the capacities.Consider an undirected graph G = (V;E). An independent set is a subset I V such that for any vertices i; j 2 I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let (G) denote the size of the largest maximal independent set in G. Consider the following greedy algorithm for generating maximal independent sets: starting with an empty set I, process the vertices in V one at a time, adding v to I is v is not connected to any vertex already in I. 2) Argue that the output I of this algorithm is a maximal independent set.
- Consider an undirected graph G = (V;E). An independent set is a subset I V such that for any vertices i; j 2 I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let (G) denote the size of the largest maximal independent set in G. 1) What is (G) if G is a complete graph on n vertices? What if G is a cycle on n vertices?Consider an undirected graph G = (V;E). An independent set is a subset I V such that for any vertices i; j 2 I, there is no edge between i and j in E. A set i is a maximal independent set if no additional vertices of V can be added to I without violating its independence. Note, however, that a maximal independent sent is not necessarily the largest independent set in G. Let (G) denote the size of the largest maximal independent set in G. One way of trying to avoid this dependence on ordering is the use of randomized algorithms. Essentially, by processing the vertices in a random order, you can potentially avoid (with high probability) any particularly bad orderings. So consider the following randomized algorithm for constructing independent sets: @ First, starting with an empty set I, add each vertex of G to I independently with probability p. @ Next, for any edges with both vertices in I, delete one of the two vertices from I (at random). @ Note - in this second step,…Let (u, v) be a directed edge in arbitrary flow network G. Prove that if there is a minimum (s, t)-cut (S, T) such that u ∈ S and v ∈ T, then there is no minimum cut (S', T' ) such that u ∈ T' , v ∈ S' ). Note that by definition of cut, s ∈ S, t ∈ T, and similarly s ∈ S' , t ∈ T'.
- Discrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2, T = A · D = A3, Q = A · T = A4. Consider the values on the diagonals. Prove that for any vertex u ∈ V with degree d(u), d(u) = Du,u.Consider a connected undirected graph G=(V,E) in which every edge e∈E has a distinct and nonnegative cost. Let T be an MST and P a shortest path from some vertex s to some other vertex t. Now suppose the cost of every edge e of G is increased by 1 and becomes ce+1. Call this new graph G′. Which of the following is true about G′ ? a) T must be an MST and P must be a shortest s - t path. b) T must be an MST but P may not be a shortest s - t path. c) T may not be an MST but P must be a shortest s - t path. d) T may not be an MST and P may not be a shortest s−t path. Pls use Kruskal's algorithm to reason about the MST.Discrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2, T = A · D = A3, Q = A · T = A4. Consider the values on the diagonals. Prove that the number of 3-cycles N3 in the whole graph is N3 = 1/6 ∑ u∈V Tu,u