Linear Assignment Problem

## Linear Sum Assignment Problem (LSAP)

All complete assignments are valid solutions. Every possible assignment has a constant rating, that is independent of other assignments:

\begin{align*} \begin{aligned} &R(x_{11}, ...,x_{NA}) = \sum\limits_{n,a}^{N,A} c_{n,a} \times x_{na}\\ &c_{na} \in \mathbb{R} \end{aligned} \end{align*}

LSAPs can be solved in $$O(N^3)$$ via the Hungarian method. The Busacker Goven algorithm can solve LSAPs in $$O(N 3 × log(N ))$$, but is often faster than the Hungarian method [DISKI 86].

## Level 1

It has the same rating function as LSAPs:

\begin{align*} \begin{aligned} &R(x_{11}, ...,x_{NA}) = \sum\limits_{n,a}^{N,A} c_{n,a} \times x_{na}\\ &c_{na} \in \mathbb{R} \end{aligned} \end{align*}

On the other hand, not all assignments valid. The validity criteria can be described via a set of linear equality and inequalities as following [DISKI 86]:

\begin{align*} \begin{aligned} &b_{11} \times x_{11} + ... + b_{1N} \times x_{1N} \leq c_1\\ &...\\ &b_{a1} \times x_{a1} + ... + b_{aN} \times x_{aN} \leq c_a\\ &d_{11} \times x_{11} + ... + d_{1N} \times x_{1N} = e_1\\ &...\\ &d_{a1} \times x_{a1} + ... + d_{aN} \times x_{aN} = e_a\\ \\ &b_{na},c_{na},d_{na},e_{na} \in \mathbb{R} \end{aligned} \end{align*}

This validity criteria only allows certain specific assignments. A linear assignment problem of the level 1 can be modeled via linear integer programming (ILP) and is therefore NP complete. Thus, this problem is not solvable with reasonable amount of resources for big instances. Instead, approximate and heuristic algorithms can be used in order to create a good enough solution.