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The knapsack problem, though NP-Hard, is one of a collection of algorithms that can still be approximated to any specified degree. This means that the problem has a polynomial time approximation scheme. To be exact, the knapsack problem has a fully polynomial time approximation scheme (FPTAS). [26]
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.
Indeed, this problem does not have an FPTAS unless P=NP. The same is true for the two-dimensional knapsack problem. The same is true for the multiple subset sum problem: the quasi-dominance relation should be: s quasi-dominates t iff max(s 1, s 2) ≤ max(t 1, t 2), but it is not preserved by transitions, by the same example as above. 2.
A practical problem with PTAS algorithms is that the exponent of the polynomial could increase dramatically as ε shrinks, for example if the runtime is O(n (1/ε)!One way of addressing this is to define the efficient polynomial-time approximation scheme or EPTAS, in which the running time is required to be O(n c) for a constant c independent of ε.
For the problem variant in which not every item must be assigned to a bin, there is a family of algorithms for solving the GAP by using a combinatorial translation of any algorithm for the knapsack problem into an approximation algorithm for the GAP. [3]
The problem is NP-hard, but it has efficient constant-factor approximation algorithms as well as an FPTAS. In practice, usually the demands s i are publicly known (e.g., the length of the advertisement of each advertiser must be known), but the valuations v i are the private information of the bidders.
NP-hard problems vary greatly in their approximability; some, such as the knapsack problem, can be approximated within a multiplicative factor +, for any fixed >, and therefore produce solutions arbitrarily close to the optimum (such a family of approximation algorithms is called a polynomial-time approximation scheme or PTAS).
The quadratic knapsack problem (QKP), first introduced in 19th century, [1] is an extension of knapsack problem that allows for quadratic terms in the objective function: Given a set of items, each with a weight, a value, and an extra profit that can be earned if two items are selected, determine the number of items to include in a collection without exceeding capacity of the knapsack, so as ...