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	~~In [[applied mathematics]], the maximum '''generalized assignment problem''' is a problem in [[combinatorial optimization]]. This problem~~ is ~~a [[generalization]] of the [[assignment problem]] in which both tasks~~ and ~~[[Agent-based model\|agents]] have a size. Moreover, the size of each task might vary from one agent to the other.~~		Claude is her name and she totally digs that name. Bookkeeping is how he supports his family and his salary has been truly fulfilling. I presently reside in Alabama. Camping is some thing that I've carried out for many years.<br><br>Feel free to surf to my web site; [http://Www.Shownetbook.com/enone/xe/?document_srl=3029968 shownetbook.com]

	~~This problem in its most general form is as follows:~~

	~~There are a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost and profit~~ that ~~may vary depending on the agent-task assignment. Moreover, each agent has a budget and the sum of the costs of tasks assigned to it cannot exceed this budget~~. It is ~~required to find an assignment in which all agents do not exceed their budget and total profit of the assignment is maximized.~~

	~~==Special cases==~~
	~~In the special case in which all the agents' budgets and all tasks' costs are equal to 1, this problem reduces to the [[maximum assignment problem]]. When the costs~~ and ~~profits of all agents-task assignment are equal, this problem reduces to the [[multiple knapsack problem]]. If there is a single agent, then, this problem reduces to the [[Knapsack problem]].~~

	~~==Definition==~~
	In the following, we have ''n'' kinds of items, <math>x_1</math> through <math>x_n</math> and ''m'' kinds of bins <math>b_1</math> through <math>b_m</math>. Each bin <math>b_i</math> is associated with a budget <math>w_i</math>. For a bin <math>b_i</math>, each item <math>x_j</math> has ~~a profit <math>p_{ij}</math> and a weight <math>w_{ij}</math>~~. ~~A solution is a subset of items ''U'' and an assignment from ''U'' to the bins. A feasible solution is a solution~~ in ~~which for each bin <math>b_i</math> the total weight of assigned items is at most <math>w_i</math>~~. ~~The solution's profit~~ is ~~the sum of profits for each item-bin assignment. The goal is to find a maximum profit feasible solution.~~

	~~Mathematically the generalized assignment problem can be formulated as:~~
	~~:maximize <math>\sum_{i=1}^m\sum_{j=1}^n p_{ij} x_{ij}.</math>~~
	~~:subject to <math>\sum_{j=1}^n w_{ij} x_{ij} \le w_i \qquad i=1, \ldots, m</math>;~~
	~~::<math> \sum_{i=1}^m x_{ij} \leq 1 \qquad j=1, \ldots, n</math>;~~
	~~::<math> x_{ij} \in \{0,1\} \qquad i=1, \ldots, m, \quad j=1, \ldots, n</math>;~~

	~~The generalized assignment problem is [[NP-hard]], and it is even [[APX-hard]] to approximate it. Recently it was shown~~ that ~~an extension of it is <math>e/(e-1) - \varepsilon</math> hard to approximate~~ for ~~every <math>\varepsilon</math>~~.~~{{Citation needed\|date=November 2012}}~~

	~~==Greedy approximation algorithm==~~
	~~Using any algorithm ALG~~ <~~math~~> ~~\alpha~~<~~/math~~>~~-approximation algorithm for the [[knapsack problem]], it is possible~~ to ~~construct a (<math> \alpha+1</math>)-approximation for the generalized assignment problem in a greedy manner using a residual profit concept.~~
	~~The algorithm constructs a schedule in iterations, where during iteration <math>j</math> a tentative selection of items~~ to ~~bin <math>b_j</math> is selected.~~
	~~The selection for bin <math>b_j</math> might change as items might be reselected in a later iteration for other bins.~~
	The residual profit of an item <math>x_i</math> for bin <math>b_j</math> is <math>p_{ij}</math> if <math>x_i</math> is not selected for any other bin or <math> p_{ij}</math> – <math>p_{ik} </math> if <math>x_i</math> is selected for bin <math>b_k</math>.

	Formally: We use a vector <math>T</math> to indicate the tentative schedule during the algorithm. Specifically, <math>T[i]=j</math> means the item <math>x_i</math> is scheduled on bin <math>b_j</math> and <math>T[i]=-1</math> means that item <math>x_i</math> is not scheduled. The residual profit in iteration <math>j</math> is denoted by <math>P_j</math>, where <math>P_j[i]=p_{ij}</math> if item <math>x_i</math> is not scheduled (i.e. <math>T[i]=-1</math>) and <math>P_j[i]=p_{ij}-p_{ik}</math> if item <math>x_i</math> is scheduled on bin <math>b_k</math> (i.e. <math>T[i]=k</math>).

	~~Formally:~~
	~~: Set <math>T[i]=-1</math> for all <math>i = 1\ldots n</math>~~
	~~: For <math>j=1...m</math> do:~~
	~~:: Call ALG to find a solution to bin <math>b_j</math> using the residual profit function <math>P_j</math>. Denote the selected items by <math>S_j</math>.~~
	~~:: Update <math>T</math> using <math>S_j</math>, i.e., <math>T[i]=j</math> for all <math>i \in S_j</math>.~~

	~~==See also==~~
	*[[Assignment problem]]

	~~== References ==~~
	* Reuven Cohen, Liran Katzir, and Danny Raz, [http://www.cs.technion.ac.il/~lirank/pubs/2006-IPL-Generalized-Assignment-Problem.pdf "An Efficient Approximation for the Generalized Assignment Problem"], Information Processing Letters, Vol. 100, Issue 4, pp. ~~162–166, November 2006.~~
	* Lisa Fleischer, Michel X. Goemans, Vahab S. Mirrokni, and Maxim Sviridenko, [http://~~www-math~~.~~mit~~.~~edu~~/~~~goemans~~/~~PAPERS~~/~~ga-soda06.pdf "Tight Approximation Algorithms for Maximum General Assignment Problems"], SODA 2006, pp. 611–620~~.
	* Hans Kellerer, Ulrich Pferschy, David Pisinger, ''Knapsack Problems '', 2005. Springer Verlag ISBN 3-540-40286-1

	~~== External links ==~~

	~~[[Category:NP-complete problems]]~~
	~~[[Category:Operations research]~~]

en>PetesGuide: /* External links */ replaced broken link to ExtremeTech "Defining scRGB" with link to same article found on PCMag.com (verified by using Wayback machine)

2012-03-10T23:30:10Z

External links: replaced broken link to ExtremeTech "Defining scRGB" with link to same article found on PCMag.com (verified by using Wayback machine)

New page

In [[applied mathematics]], the maximum '''generalized assignment problem''' is a problem in [[combinatorial optimization]]. This problem is a [[generalization]] of the [[assignment problem]] in which both tasks and [[Agent-based model|agents]] have a size. Moreover, the size of each task might vary from one agent to the other.

This problem in its most general form is as follows:

There are a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost and profit that may vary depending on the agent-task assignment. Moreover, each agent has a budget and the sum of the costs of tasks assigned to it cannot exceed this budget. It is required to find an assignment in which all agents do not exceed their budget and total profit of the assignment is maximized.

==Special cases==
In the special case in which all the agents' budgets and all tasks' costs are equal to 1, this problem reduces to the [[maximum assignment problem]]. When the costs and profits of all agents-task assignment are equal, this problem reduces to the [[multiple knapsack problem]]. If there is a single agent, then, this problem reduces to the [[Knapsack problem]].

==Definition==
In the following, we have ''n'' kinds of items, <math>x_1</math> through <math>x_n</math> and ''m'' kinds of bins <math>b_1</math> through <math>b_m</math>. Each bin <math>b_i</math> is associated with a budget <math>w_i</math>. For a bin <math>b_i</math>, each item <math>x_j</math> has a profit <math>p_{ij}</math> and a weight <math>w_{ij}</math>. A solution is a subset of items ''U'' and an assignment from ''U'' to the bins. A feasible solution is a solution in which for each bin <math>b_i</math> the total weight of assigned items is at most <math>w_i</math>. The solution's profit is the sum of profits for each item-bin assignment. The goal is to find a maximum profit feasible solution.

Mathematically the generalized assignment problem can be formulated as:
:maximize <math>\sum_{i=1}^m\sum_{j=1}^n p_{ij} x_{ij}.</math>
:subject to <math>\sum_{j=1}^n w_{ij} x_{ij} \le w_i \qquad i=1, \ldots, m</math>;
::<math> \sum_{i=1}^m x_{ij} \leq 1 \qquad j=1, \ldots, n</math>;
::<math> x_{ij} \in \{0,1\} \qquad i=1, \ldots, m, \quad j=1, \ldots, n</math>;

The generalized assignment problem is [[NP-hard]], and it is even [[APX-hard]] to approximate it. Recently it was shown that an extension of it is <math>e/(e-1) - \varepsilon</math> hard to approximate for every <math>\varepsilon</math>.{{Citation needed|date=November 2012}}

==Greedy approximation algorithm==
Using any algorithm ALG <math> \alpha</math>-approximation algorithm for the [[knapsack problem]], it is possible to construct a (<math> \alpha+1</math>)-approximation for the generalized assignment problem in a greedy manner using a residual profit concept.
The algorithm constructs a schedule in iterations, where during iteration <math>j</math> a tentative selection of items to bin <math>b_j</math> is selected.
The selection for bin <math>b_j</math> might change as items might be reselected in a later iteration for other bins.
The residual profit of an item <math>x_i</math> for bin <math>b_j</math> is <math>p_{ij}</math> if <math>x_i</math> is not selected for any other bin or <math> p_{ij}</math> – <math>p_{ik} </math> if <math>x_i</math> is selected for bin <math>b_k</math>.

Formally: We use a vector <math>T</math> to indicate the tentative schedule during the algorithm. Specifically, <math>T[i]=j</math> means the item <math>x_i</math> is scheduled on bin <math>b_j</math> and <math>T[i]=-1</math> means that item <math>x_i</math> is not scheduled. The residual profit in iteration <math>j</math> is denoted by <math>P_j</math>, where <math>P_j[i]=p_{ij}</math> if item <math>x_i</math> is not scheduled (i.e. <math>T[i]=-1</math>) and <math>P_j[i]=p_{ij}-p_{ik}</math> if item <math>x_i</math> is scheduled on bin <math>b_k</math> (i.e. <math>T[i]=k</math>).

Formally:
: Set <math>T[i]=-1</math> for all <math>i = 1\ldots n</math>
: For <math>j=1...m</math> do:
:: Call ALG to find a solution to bin <math>b_j</math> using the residual profit function <math>P_j</math>. Denote the selected items by <math>S_j</math>.
:: Update <math>T</math> using <math>S_j</math>, i.e., <math>T[i]=j</math> for all <math>i \in S_j</math>.

==See also==
*[[Assignment problem]]

== References ==
* Reuven Cohen, Liran Katzir, and Danny Raz, [http://www.cs.technion.ac.il/~lirank/pubs/2006-IPL-Generalized-Assignment-Problem.pdf "An Efficient Approximation for the Generalized Assignment Problem"], Information Processing Letters, Vol. 100, Issue 4, pp. 162–166, November 2006.
* Lisa Fleischer, Michel X. Goemans, Vahab S. Mirrokni, and Maxim Sviridenko, [http://www-math.mit.edu/~goemans/PAPERS/ga-soda06.pdf "Tight Approximation Algorithms for Maximum General Assignment Problems"], SODA 2006, pp. 611–620.
* Hans Kellerer, Ulrich Pferschy, David Pisinger, ''Knapsack Problems '', 2005. Springer Verlag ISBN 3-540-40286-1

== External links ==

[[Category:NP-complete problems]]
[[Category:Operations research]]