# Algorithm 算法作业代写：高质量原创代码，80分以下退款

##### 如何设计一个算法？

3个数字相加并打印其总和的算法。

1. a) Consider the following bipartite graph G = (V1, V2, E)

Determine if G features

i) an Eulerian circuit, and

ii) a Hamiltonian cycle.

In both cases state your reasoning and, if possible, give your solutions.

b) Using the graph from Part a), let M = {{v1, v7}, {v2, v8}, {v3, v9}} be a maximal matching on G. Now use M to determine a maximum matching on G using the augmenting path algorithm, showing your steps.

c) Show how the maximum matching problem on bipartite graphs can also be solved using the Ford-Fulkerson algorithm.

d) A protest group wants to prevent a politician from attending parliament by blocking all possible driving routes between their home and the parliament building. They would like to do this while blocking the minimum possible number of roads. Discuss how this subset of roads can be determined.

2、a) Given a simple graph, give definitions and draw examples of the following:

i) a bridge,

ii) a cut vertex,

iii) a complete graph,

iv) a complement graph,

v) a simple path.

b) Given a simple graph G = (V, E), let L(v) denote a list of “permissible” colours for each vertex v in V. We are now interested in colouring the vertices of V such that

• Adjacent vertices are always assigned to different colours
• The number of different colours used across the graph is minimal
• Each vertex v is assigned to a colour appearing in its list L(v).

Prove that this problem is NP-complete / NP-hard.

c) Design a simple greedy algorithm for the problem stated in Part b). In cases where a vertex cannot be coloured, it is acceptable to leave it uncoloured.

d) Consider the problem of sorting a list of n integers. It is easy to check (in O(n) time)

whether any arbitrary arrangement is in numerical order. On the other hand, the number of different ways of arranging n integers is O(n!). Does this mean that the sorting problem is NP-complete? In your answer, fully discuss your reasoning.

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