Linear bottleneck assignment problem

Key takeaways

Linear bottleneck assignment problem belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Linear bottleneck assignment problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Linear bottleneck assignment problem from memory before moving on to harder problems.

Reference excerpt

In combinatorial optimization, a field within mathematics, the linear bottleneck assignment problem (LBAP) is similar to the linear assignment problem. In plain words the problem is stated 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 the agent-task assignment. It is required to perform all tasks by assigning exactly one agent to each task in such a way that the maximum cost among the individual assignments is minimized. The term "bottleneck" is explained by a common type of application of the problem, where the cost is the duration of the task performed by an agent. In this setting the "maximum cost" is "maximum duration", which is the bottleneck for the schedule of the overall job, to be minimized.

Formal definition The formal definition of the bottleneck assignment problem is

Given two sets, A and T, together with a weight function C : A × T → R. Find a bijection f : A → T such that the cost function:

max a ∈ A C ( a , f ( a ) ) {\displaystyle \max _{a\in A}C(a,f(a))}

is minimized. Usually the weight function is viewed as a square real-valued matrix C, so that the cost function is written down as:

max a ∈ A C a , f ( a ) {\displaystyle \max _{a\in A}C_{a,f(a)}}

Mathematical programming formulation

min max i , j c i j x i j {\displaystyle \min \,\max _{i,j}c_{ij}x_{ij}}

subject to:

∑ j = 1 n x i j = 1 ( i = 1 , 2 , … , n ) , {\displaystyle \sum _{j=1}^{n}x_{ij}=1(i=1,2,\dots ,n),}

∑ i = 1 n x i j = 1 ( j = 1 , 2 , … , n ) , {\displaystyle \sum _{i=1}^{n}x_{ij}=1(j=1,2,\dots ,n),}

x i j ∈ { 0 , 1 } ( i , j = 1 , 2 , … , n ) {\displaystyle x_{ij}\in \{0,1\}(i,j=1,2,\dots ,n)}

Asymptotics Let c n ∗ {\displaystyle c_{n}^{*}} denote the optimal objective function value for the problem with n agents and n tasks. If the costs c i j {\displaystyle c_{ij}} are sampled from the uniform distribution on (0,1), then

E [ c n ∗ ] = log ⁡ n + log ⁡ 2 + γ n + O ( ( log ⁡ n ) 2 n 7 / 5 ) {\displaystyle E[c_{n}^{*}]={\frac {\log n+\log 2+\gamma }{n}}+O\left({\frac {(\log n)^{2}}{n^{7/5}}}\right)}

and

V a r [ c n ∗ ] = ζ ( 2 ) − 2 ( log ⁡ 2 ) 2 n 2 + O ( ( log ⁡ n ) 2 n 7 / 3 ) . {\displaystyle Var[c_{n}^{*}]={\frac {\zeta (2)-2(\log 2)^{2}}{n^{2}}}+O\left({\frac {(\log n)^{2}}{n^{7/3}}}\right).}

References

Worked examples

Example 1 — a first encounter with Linear bottleneck assignment problem

Start with the simplest possible case. Write down what Linear bottleneck assignment problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Linear bottleneck assignment problem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Linear bottleneck assignment problem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Linear bottleneck assignment problem

In research: Linear bottleneck assignment problem appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Linear bottleneck assignment problem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Linear bottleneck assignment problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Combinatorial optimization, so understanding it makes those chapters shorter.
In everyday life: Look for Linear bottleneck assignment problem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Linear bottleneck assignment problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Linear bottleneck assignment problem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Linear bottleneck assignment problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Linear bottleneck assignment problem in simple terms?

In combinatorial optimization, a field within mathematics, the linear bottleneck assignment problem (LBAP) is similar to the linear assignment problem. In plain words the problem is stated as follows: There are a number of agents and a number of tasks.

Why does Linear bottleneck assignment problem matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Linear bottleneck assignment problem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Linear bottleneck assignment problem.