The assignment problem is specified by the cost matrix cij for minimization problems or the benefit matrix aij for maximization problems.Those matrices describe the cost or benefit of assigning object j to person i.
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Visit Stack Exchange I wonder if there is any literature on the following problem $$\begin \underset & \displaystyle\sum_ C_ X_\\ \text & \displaystyle\sum_ X_ = \displaystyle\sum_ X_ = 1\\ & X_ \geq 0\end$$ The closest related problem might be Rectangular Linear Assignment Problem (RLAP)$^\dagger$, as RLAP further constrains $X_ \in \$.
The LSAP is also used as subproblems in a number of network flow and applied combinatorial optimization problems such as the quadratic assignment problem (QAP), the traveling salesman problem (TSP) and the transportation problem.
The assignment problem is also used as a preprocessing step for pivoting methods.
So, do the relaxed RLAP share the same optimum as RLAP? The converted problem is a standard assignment problem whose optimal objective cost is the same as that of the Rectangular Assignment Problem.
Per the Integrality Theorem, the Assignment Problem, without integrality constraints, has an optimal solution consisting of all integers (which must be 0 and 1).Due to the dynamic nature of many applications, as well as the expansion of problem sizes, where a solution needs to be found under tight time constraints, heuristics that give rise to solutions that are close to optimal solution are sought.Our efforts in that regards lead us to the Deep Greedy Switching (DGS) algorithm.The auction algorithm proved to be unsuitable for our application.Even attempts of a parallelized version of the auction algorithm did not meet our requirements.Even as the auction is considered one of the fastest algorithms, for large-scale and complex instances of the assignment problem the auction algorithm can take a lot of time to find the optimal solution. discussed five types of the assignment problem instances which are Geometric, Fixed-Cost, High-Cost, Low-Cost and Uniformly Random.It was shown that for the first two problem types the auction algorithm performs poorly in terms of the running time in comparison with the other three problem types.Trick applied a greedy heuristic approach for the generalized assignment problem, where it was shown that some randomization to the greedy approach is required to reach better results than a completely greedy approach.Another heuristic approach is the greedy randomized adaptive search procedure (GRASP, which consist of a multi-start random initial solutions and a local search.approximate dual projective algorithm by Ramakrishnan et al.) and algorithms that use forest construction as dual forest algorithm by Achatz et al.The auction algorithm is considered one of the fastest algorithm that can find the optimal solution for the assignment problem.