Course Notes - Mathematics
... Problem 1.1.9 (Labeled Triangles) Draw a triangle T and label its vertices 1, 2, and 3. Subdivide the triangle into smaller triangles, introducing new vertices if you wish, which can also be along the edges of T . Label each new vertex 1, 2, or 3, any way you want, with the following restriction: Yo ...
... Problem 1.1.9 (Labeled Triangles) Draw a triangle T and label its vertices 1, 2, and 3. Subdivide the triangle into smaller triangles, introducing new vertices if you wish, which can also be along the edges of T . Label each new vertex 1, 2, or 3, any way you want, with the following restriction: Yo ...
Guided Local Search Joins the Elite in Discrete Optimisation 1
... methods, the basic form of which are often referred to as hill-climbing. To perform hill-climbing, one must define the following: (a) a representation for candidate solutions; (b) an objective function: given any candidate solution, this function returns a numerical value. The problem is seen as an ...
... methods, the basic form of which are often referred to as hill-climbing. To perform hill-climbing, one must define the following: (a) a representation for candidate solutions; (b) an objective function: given any candidate solution, this function returns a numerical value. The problem is seen as an ...
Single Buyer, Single Seller Auction
... Mechanisms to be useful and provide better approximation guarantee, we need to answer the following two questions: Question 1. What is the worst gap between the revenue obtained by optimal partition mechanism and optimal mechanism? Is it less than 6? Question 2. Can we find optimal/approximately-opt ...
... Mechanisms to be useful and provide better approximation guarantee, we need to answer the following two questions: Question 1. What is the worst gap between the revenue obtained by optimal partition mechanism and optimal mechanism? Is it less than 6? Question 2. Can we find optimal/approximately-opt ...
50 MATHCOUNTS LECTURES (24)
... Example 2. In a group of 2 cats, 3 dogs, and 10 pigs in how many ways can we choose a committee of 6 animals if (a) there are no constraints in species? (b) the two cats must be included? (c) the two cats must be excluded? (d) there must be at least 3 pigs? (e) there must be at most 2 pigs? (f) Joe ...
... Example 2. In a group of 2 cats, 3 dogs, and 10 pigs in how many ways can we choose a committee of 6 animals if (a) there are no constraints in species? (b) the two cats must be included? (c) the two cats must be excluded? (d) there must be at least 3 pigs? (e) there must be at most 2 pigs? (f) Joe ...
Knapsack problem
The knapsack problem or rucksack problem is a problem in combinatorial optimization: Given a set of items, each with a mass and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items.The problem often arises in resource allocation where there are financial constraints and is studied in fields such as combinatorics, computer science, complexity theory, cryptography and applied mathematics.The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. It is not known how the name ""knapsack problem"" originated, though the problem was referred to as such in the early works of mathematician Tobias Dantzig (1884–1956), suggesting that the name could have existed in folklore before a mathematical problem had been fully defined.