Let G be a graph with 2n vertices, and assume either deg v ≥ n for each vertex v, or |E| ≥ 1 2 (2n − 1)(2n − 2) + 2. Show that G has a perfect matching. Hint: Derive these assertions from a more general result involving Hamiltonian cycles. Example 13.2.1 illustrates the following […]

Let G be a connected graph, and assume that every matching in G can be extended to a perfect matching; such a graph is called randomly matchable. Prove that the only randomly matchable graphs on 2n vertices are the graphs Kn,n and K2n; see [Sum79] and [LePP84]. Hint: Show first that G has to be 2-connected. If G is bipartite and contains non-adjacent vertices s and t which are in different parts of G,

Use the method described in Example 13.4.1 to enlarge the matching shown in the graph of Figure 13.9. Take r as the root of the alternating tree; if choices have to be made, use the vertices according to increasing labels. Hint: You can simplify this task by exploiting the inherent symmetry of the graph in question, which

Show that the bottleneck assignment problem defined in Example 7.4.11 is a special case of the algebraic assignment problem; see [Law76, §5.7] and [GaTa88] for that problem. Is every optimal matching also product-optimal? Describe the problem of finding an optimal integral circulation on a network (G, b, c) as an ILP. Also, describe the problem of

Determine a product-optimal matching for the graph with respect to the weight matrix of Example 14.2.5; that is, we seek a perfect matching for which the product of the weights of its edges is maximal. Hint: Apply the Hungarian algorithm within the group (Q+, ・); note that the zero of this group is 1, and

Use various sources (including Prudential Retirement Planning) and course-related concepts to examine your current financial condition and life goals. Apply this knowledge to develop suitable retirement and estate plans. Your contingency plan should incorporate sufficient detail to demonstrate both course knowledge and financial sophistication. Prepare a three- to six-page paper (excluding the title and reference pages) in

Discuss how deforestation of tropical rainforests in Liberia poses a threat to the rich biological diversity of tropical rainforests, hinders the advancement of medicinal discoveries, and contributes largely to the greenhouse effect. Deforestation of tropical rainforests in Liberia:Medicinal discoveries and greenhouse effect It needs to follow my topic which is “Deforestation of tropical rainforests in

For the TSP of Example 15.1.2, we obtain the minimal spanning tree T with w(T) = 186 shown in Figure 15.2. This bound is slightly inferior to the one provided in Example 15.2.1, but determining a minimal spanning tree is also much easier than solving an AP. Note that the MST relaxation leaves out the

We choose s = Be in Example 15.1.2; this choice is motivated by the fact that the sum of the two smallest edge weights is maximal for this vertex. We obtain the s-tree B shown in Figure 15.3; note that B is the minimal spanning tree T given in Figure 15.2, with the edge BeNu added. Hence w(TSP) ≥ w(B) = 186 + 44 = 230.

Determine a minimal s-tree for the TSP of Example 15.1.2 for the other possible choices of the special vertex s. Determine the number of s-trees of . Hint: Use Corollary 1.2.11. Show that replacing W by W” according to the definition of pin (15.8) does not change the weight of a tour. There remains the problem of choosing the value of c. It is possible to