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  2. Continuous knapsack problem - Wikipedia

    en.wikipedia.org/wiki/Continuous_knapsack_problem

    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.

  3. Knapsack problem - Wikipedia

    en.wikipedia.org/wiki/Knapsack_problem

    The most common problem being solved is the 0-1 knapsack problem, which restricts the number of copies of each kind of item to zero or one. Given a set of n {\displaystyle n} items numbered from 1 up to n {\displaystyle n} , each with a weight w i {\displaystyle w_{i}} and a value v i {\displaystyle v_{i}} , along with a maximum weight capacity ...

  4. List of knapsack problems - Wikipedia

    en.wikipedia.org/wiki/List_of_knapsack_problems

    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.

  5. Packing problems - Wikipedia

    en.wikipedia.org/wiki/Packing_problems

    Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap. In a bin packing problem, people are given: A container, usually a two- or three-dimensional convex region, possibly of infinite size. Multiple containers ...

  6. Bin packing problem - Wikipedia

    en.wikipedia.org/wiki/Bin_packing_problem

    The problem of fractional knapsack with penalties was introduced by Malaguti, Monaci, Paronuzzi and Pferschy. [44] They developed an FPTAS and a dynamic program for the problem, and they showed an extensive computational study comparing the performance of their models.

  7. Karmarkar–Karp bin packing algorithms - Wikipedia

    en.wikipedia.org/wiki/Karmarkar–Karp_bin...

    The knapsack problem can be solved by dynamic programming in pseudo-polynomial time: (), where m is the number of inputs and V is the number of different possible values. To get a polynomial-time algorithm, we can solve the knapsack problem approximately, using input rounding.

  8. Configuration linear program - Wikipedia

    en.wikipedia.org/wiki/Configuration_linear_program

    The separation oracle for the dual LP can be implemented by solving the knapsack problem with sizes s and values y: if the optimal solution of the knapsack problem has a total value at most 1, then y is feasible; if it is larger than 1, than y is not feasible, and the optimal solution of the knapsack problem identifies a configuration for which ...

  9. Combinatorial participatory budgeting - Wikipedia

    en.wikipedia.org/wiki/Combinatorial...

    The proof is by reduction from the knapsack problem. Given a knapsack problem, define a PB instance with a single voter in which the budget is the knapsack capacity, and for each item with weight w and value v, there is a project with cost w and utility v. Let P be the optimal solution to the knapsack