# Questions tagged [assignment-problem]

For questions about the assignment problem in combinatorial optimization, NOT for problems that you've been set as a homework assignment.

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### Assignment problem for multiple days

I have a problem that can be reduced to an assignment problem. (In a previous question i found out how to do that.) Which means we have a set $A$ of agents and a set $T$ of tasks as well as a cost ...
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### XOR-like behavior in flow networks

XOR is not the correct name, but I am looking for some kind of exclusive behavior. I am currently solving a set of different (assignment) problems by modeling flow networks and running a min-cost-max-...
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### CNF form of variable assignment problem

There are n variables $x_1$, $x_2$,..., $x_n$ and each one of them takes values from 1 to k (k>= n) and all are distinct. How can I represent this in the CNF form? (I tried the trivial way of trying ...
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### A variant of the Assignment Problem

In my variant of the assignment problem I have a set $A$ of agents and a set (of possibly different cardinality) $T$ of tasks. Each agent needs to be assigned exactly $n$ or $n+1$ tasks, and each task ...
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### Maximizing the sum of selected elements in a matrix

I’m trying to find an efficient algorithm for the following optimization problem: Given a matrix $A$ with elements $a_{ij}$ and dimension $k$, select exactly $n$ elements from $A$ ($n<k$) such ...
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### Heuristic algorithms for the dense assignment problem

Given a dense assignment problem ($n$ tasks assigned to $n$ workers, where each worker can do any one of the tasks), I understand the best complexity is $O(n^3)$, using the Hungarian Algorithm or ...
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I have this simple 'assignment' problem: We have a set of agents $A = \{a_1, a_2, \dotso, a_n\}$ and set of tasks $T= \{t_1, t_2, \dotso, t_m\}$. Note that $m$ is not necessarily equal to $n$. Unlike ...
243 views

### Maximum flow on a tripartite graph

I have to solve an assignment problem between $\{1,\dots, N\}$ agents and $\{1,\dots, M\}$ objects, which comes to maximize : \begin{equation} \sum_{ij}\beta_{ij}x_{ij} \end{equation} where $x_{ij}$ ...
219 views

### Discrete assignment problem with penalties

I came across a problem were you have to plan an optimal assignment pattern. Let's say you have $j=1,\ldots,n$ tasks during $i=1,\ldots,m$ time periods. It's an single agent problem where we have to ...
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### Working Optimization Algorithm

This is my basic setup: I have five time slots (A, B, C, D, E; each day has these five time slots) over 60 days that all need to be filled with n people each (n must remain a variable). However, each ...
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### Stable marriage problem with only one side having preferences [duplicate]

I was wondering about a variation on the Stable Marriage Problem. Initially, we have two sets of entities, usually males and females, and they have preference lists ranking the other group, and ...
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### Minimizing total distance traveled by points in points cloud transformation

I have a point cloud of size n (in the example on picture n = 5). The starting coordinates are in green, destination coordinates are in red. What I need is to move the points from the starting ...
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### How do solve assignment of one interval to another?

Is there an efficient algorithm for the following problem? Input: Set of holes and pegs. Each hole/peg is an interval $[\ell,u]$ with integer endpoints. Question: Can all the holes be filled ... 426 views

### Assignment problem with no cost

I have a problem that I was able to conceptualize as following: Problem We have a set of n people. And m subsets representing their ethnicity like White, Hispanic, Asian etc. Given any combination of ...
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### Assign n people to m rooms of different sizes, such that noone is alone

I'm looking for an efficient way to assign n people to m rooms in a very specific way. INPUT: The program receives two sets of people (set of males and set of females), as well as a set of available ...
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### Assignment Problem -- finding the $k$ agents with the best assignment

I have a question that I have been thinking about. Suppose we have $n$ agents, $m$ tasks, a cost matrix with $M_{ij}$ being the cost of agent $i$ performing task $j$, and are given a value $k \leq n$. ...
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I want to assign $m$ tasks to $n$ workers where $m>n$, so as to minimize assignment costs defined by an $m \times n$ matrix $C$. That is, I want to find Boolean variables $x_{i,j}$ which minimize ...
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### affinity based static load balancing

I am trying to find a good model the following problem: Given a collection of work packets x, y, z, ..., and a collection of worker nodes ...
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### Optimality in multi-agent multi-target path finding

Suppose I have a regular rectangular weighted grid with multiple agents and obstacles. Agents cannot be in grid sites that contain obstacles, and for simplicity assume multiple agents can be in the ...
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### Algorithm for a list of best solutions to the Assignment problem

I am trying to brute force a classical substitution cipher. The problem is that there are $26!$ possible keys. So, I'd like to do frequency analysis to try likely keys first. Then, on the first $n$ ... 1k views

### What is the appropriate algorithm for bipartite matching with constraints?

I have a problem that is a bit complex, and I don't know what method/model I should use to express it (much less solve it). Let's say we have a lot of employees and a few jobs to be done. Each ...
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### A variant of job assignment (scheduling) problem with variable time span

The problem is a scheduling problem with n jobs and k machines. Each job i can be started at any time, but its duration is not exactly known except a time span interval. For example, a job may take ...
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### Moving objects onto a line segment so that their pairwise similarities are close to their pairwise distances on the line

I have some arbitrary pairwise similarity metric for some objects, and I am considering trying to find the best way to position the objects onto a line segment such that the pairwise euclidean ...
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### How can I solve this constrained assignment problem?

The assignment problem is defined 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 that may vary depending on ...
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### Seating arrangement problem

$n$ professors go to a conference and have to sit together at a table. See illustration below for $n=8$ Each professor has people they like to sit next to and people they do not want to sit next ...
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### How to solve this very complicated assignment problem

A set of m items need to be placed into n stacks, where m > n. Each stack has z positions. An item has different widths when placed into different positions in a stack. The width of an item depends ...
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### Polynomial time solution for bipartite matching

Inspired by this StackOverflow question, I am wondering if there is an efficient algorithm for the following problem: Assume $n$ items and $n$ boxes, with all boxes numbered numerically and all ...
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### Choice of algorithm for hierarchical clustering for minimizing network communication costs

Suppose I have a large distributed task running on a cluster system where part of the workload is compute bound and part depends on network performance. Data transfer is not completely homogeneous ...
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### Assignment problem or something else?

My primary goal with this question is to identify the type of problem I have so I know what solution to pursue. I think it's either an instance of the assignment problem or the fair item allocation ...
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### Bipartite Perfect Matching "Assignment Problem" - finding an assignment of a particular weight

The assignment problem is to find the minimum weight perfect matching in a weighted bipartite graph. This problem can be solved using the Hungarian algorithm in polynomial time. It is also possible to ...
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### Finding a simple algorithm for an assignment problem

I have an assignment problem where I need to assign resources to consumers. Each consumer has a list of resources it'd like to acquire, ordered by preference, and each consumer can have at most one ...
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### Min Cost Max Flow algorithms for providing multiple solutions

Minimum Cost Maximum Flow algorithms have been known to provide an optimal flow routing for network flow problems in satisfactory runtime. Some of the algorithms for solving a min-cost max-flow ...
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### Assignment Problem with Minimum and Maximum constraints [duplicate]

I have the following problem: In a school, there are n students and m clubs, with n > m. Each student needs to be assigned a club. The students have preferences, (say top 3 or top 5) of the clubs ...
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### Assign $m$ tasks to $n$ workers, with $m \geq n$

There are $n$ students that share the same apartment. At each evening, one of them must prepare dinner for everyone. There are $m$ evenings to schedule, with $m \geq n$, and you have to assign any ...
We're given a set of items $A=\{1,\dots,m\}$ and a set of people $B=\{1,\dots,n\}$. Each person has a preference ordering for the items in $A$. Each item in $A$ has a specific positive cost for each ...