Chapter 2: Business Efficiency
What is the formula for the number of unique Hamiltonian circuits for a complete graph with n vertices?
(n-1)! / 2
What is a minimum-cost Hamiltonian circuit?
A Hamiltonian circuit in a graph with weights on the edges, for which the sum of the weights of the edges of the Hamiltonian circuit is as small as possible.
What is a Hamiltonian circuit?
A circuit using distinct edges of a graph that starts and ends at a particular vertex of the graph and visits each vertex once and only once. A Hamiltonian circuit can start at any one of its vertices.
What are NP-complete problems?
A collection of problems, which includes the TSP, that appear to be very hard to solve quickly for an optimal solution.
What is a tree?
A connected graph with no circuits.
What is an order-requirement digraph?
A directed graph that shows which tasks precede other tasks among the collection of tasks making up a job.
What is a complete graph?
A graph in which each pair of vertices is joined by an edge.
What is the Fundamental Principle of Counting?
A method for counting the outcomes of multistage processes. If there are a ways of choosing one thing, b ways of choosing a second after the first, ..., and z ways of choosing the last item after the earlier choices, then the total number of choice patterns is a*b*...*z.
What is a heuristic algorithm?
A method of solving an optimization problem that is "fast" but does not guarantee an optimal answer to the problem.
What is weight?
A number assigned to an edge of a graph that can be thought of as the cost, distance, or time associated with that edge.
What is a minimum-cost spanning tree?
A spanning tree of a weighted connected graph having minimum cost. The cost of a tree is the sum of the weights on the edges of the tree.
What is an algorithm?
A step-by-step description of how to solve a problem.
What is a spanning tree?
A subgraph of a connected graph that is a tree and includes all the vertices of the original graph.
What is the method of trees?
A visual method of carrying out the fundamental principle of counting.
What is Kruskal's algorithm?
An algorithm developed by Joseph Kruskal that solves the minimum-cost spanning-tree problem by selecting edges in order of increasing cost, but in such a way that no edge forms a circuit with edges chosen earlier. It can be proved that this algorithm always produces an optimal solution.
What is the nearest-neighbor algorithm?
An algorithm for attempting to solve the TSP that begins at a "home" vertex and visits next the vertex not already visited that can be reached most cheaply. When all other vertices have been visited, the tour returns to home. This method may not give an optimal answer.
What is the sorted-edges algorithm?
An algorithm for attempting to solve the TSP where the edges added to the circuit being built up are selected in order of increasing cost, but no edge is chosen that would prevent a Hamiltonian circuit from forming. The edges must all be connected at the end, but not necessarily at earlier stages. The tour obtained may not have the lowest possible cost.
What is the greedy algorithm?
An approach for solving an optimization problem, where at each stage of the algorithm the best (or cheapest) action is taken. Unfortunately, greedy algorithms do not always lead to optimal solutions.
What are the steps to finding a minimum-cost Hamiltonian circuit?
Generate all possible Hamiltonian tours. Add up the distances on the edges of each tour. Choose a tour with the smallest total distance.
Is there a simple method for determining whether or not a graph has a Hamiltonian circuit?
No.
What is a critical path?
The longest path in an order-requirement digraph. The length of this path gives the earliest completion time for all the tasks making up the job consisting of the tasks in a digraph.
What is the brute force method?
The method that solves the traveling salesman problem (TSP) by enumerating all Hamiltonian circuits and then selecting the one with minimum cost.
What is the Traveling salesman problem (TSP)?
The problem of finding a minimum-cost Hamiltonian circuit in a complete graph where each edge has been assigned a cost (or weight).