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This is an unbalanced assignment problem. One way to solve it is to invent a fourth dummy task, perhaps called "sitting still doing nothing", with a cost of 0 for the taxi assigned to it. This reduces the problem to a balanced assignment problem, which can then be solved in the usual way and still give the best solution to the problem.
The Hungarian method is a combinatorial optimization algorithm that solves the assignment problem in polynomial time and which anticipated later primal–dual methods.It was developed and published in 1955 by Harold Kuhn, who gave it the name "Hungarian method" because the algorithm was largely based on the earlier works of two Hungarian mathematicians, Dénes Kőnig and Jenő Egerváry.
The quadratic assignment problem (QAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics, from the category of the facilities location problems first introduced by Koopmans and Beckmann. [1] The problem models the following real-life problem:
The general regression model with n observations and k explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is = + where y is an n × 1 vector of dependent variable observations, each column of the n × k matrix X is a vector of observations on one of the k explanators, is a k × 1 vector of true coefficients, and e is an n × 1 vector of the ...
A solution is an assignment from items to bins. A feasible solution is a solution in which for each bin the total weight of assigned items is at most . The solution's profit is the sum of profits for each item-bin assignment. The goal is to find a maximum profit feasible solution.
While it is easy to verify whether a given assignment renders the formula true, [1] no essentially faster method to find a satisfying assignment is known than to try all assignments in succession. Cook and Levin proved that each easy-to-verify problem can be solved as fast as SAT, which is hence NP-complete.
PSM has been shown to increase model "imbalance, inefficiency, model dependence, and bias," which is not the case with most other matching methods. [3] The insights behind the use of matching still hold but should be applied with other matching methods; propensity scores also have other productive uses in weighting and doubly robust estimation.
A computationally difficult variation of 2-satisfiability, finding a truth assignment that maximizes the number of satisfied constraints, has an approximation algorithm whose optimality depends on the unique games conjecture, and another difficult variation, finding a satisfying assignment minimizing the number of true variables, is an ...