Search results
Results From The WOW.Com Content Network
The knapsack problem is one of the most studied problems in combinatorial optimization, with many real-life applications. For this reason, many special cases and generalizations have been examined. For this reason, many special cases and generalizations have been examined.
The DP works in n steps. At each step i, it processes the input x i, and constructs a set of states S i. Each state encodes a partial solution to the problem, using inputs x 1,...,x i. The components of the DP are: A set S 0 of initial states. A set F of transition functions. Each function f in F maps a pair (state,input) to a new state.
The knapsack problem is interesting from the perspective of computer science for many reasons: The decision problem form of the knapsack problem (Can a value of at least V be achieved without exceeding the weight W?) is NP-complete, thus there is no known algorithm that is both correct and fast (polynomial-time) in all cases.
For the one-dimensional case, the new patterns are introduced by solving an auxiliary optimization problem called the knapsack problem, using dual variable information from the linear program. The knapsack problem has well-known methods to solve it, such as branch and bound and dynamic programming. The Delayed Column Generation method can be ...
In theoretical computer science, the continuous knapsack problem (also known as the fractional knapsack problem) is an algorithmic problem in combinatorial optimization in which the goal is to fill a container (the "knapsack") with fractional amounts of different materials chosen to maximize the value of the selected materials.
It is a special case of the integer knapsack problem, and has applications wider than just currency. It is also the most common variation of the coin change problem , a general case of partition in which, given the available denominations of an infinite set of coins, the objective is to find out the number of possible ways of making a change ...
For example, the NP-hard knapsack problem can be solved by a dynamic programming algorithm requiring a number of steps polynomial in the size of the knapsack and the number of items (assuming that all data are scaled to be integers); however, the runtime of this algorithm is exponential time since the input sizes of the objects and knapsack are ...
The hexagonal packing of circles on a 2-dimensional Euclidean plane. These problems are mathematically distinct from the ideas in the circle packing theorem.The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere.