computer science

Hardness of approximation

Hardness of approximation is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hardness of approximation rather than just read about it. In short: In computer science, hardness of approximation is a field that studies the algorithmic complexity of finding near-optimal solutions to optimization problems. Scope Hardness of approximation complements the study of approximation algorithms by proving, for certain problems, a limit on the factors with which their solution can be efficiently approximated.

Key takeaways

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

Reference excerpt

In computer science, hardness of approximation is a field that studies the algorithmic complexity of finding near-optimal solutions to optimization problems.

Scope Hardness of approximation complements the study of approximation algorithms by proving, for certain problems, a limit on the factors with which their solution can be efficiently approximated. Typically such limits show a factor of approximation beyond which a problem becomes NP-hard, implying that finding a polynomial time approximation for the problem is impossible unless NP=P. Some hardness of approximation results, however, are based on other hypotheses, a notable one among which is the unique games conjecture.

History Since the early 1970s it was known that many optimization problems could not be solved in polynomial time unless P = NP, but in many of these problems the optimal solution could be efficiently approximated to a certain degree. In the 1970s, Teofilo F. Gonzalez and Sartaj Sahni began the study of hardness of approximation, by showing that certain optimization problems were NP-hard even to approximate to within a given approximation ratio. That is, for these problems, there is a threshold such that any polynomial-time approximation with approximation ratio beyond this threshold could be used to solve NP-complete problems in polynomial time. In the early 1990s, with the development of PCP theory, it became clear that many more approximation problems were hard to approximate, and that (unless P = NP) many known approximation algorithms achieved the best possible approximation ratio. Hardness of approximation theory deals with studying the approximation threshold of such problems.

Examples For an example of an NP-hard optimization problem that is hard to approximate, see set cover and vertex cover.

Worked examples

Example 1 — a first encounter with Hardness of approximation

Start with the simplest possible case. Write down what Hardness of approximation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer 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 Hardness of approximation 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 Hardness of approximation 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 Hardness of approximation

In research: Hardness of approximation appears in computer 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 Hardness of approximation 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: Hardness of approximation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Approximation algorithms, Computational complexity theory, Relaxation (approximation), so understanding it makes those chapters shorter.
In everyday life: Look for Hardness of approximation 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.

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How to study Hardness of approximation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hardness of approximation 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 Hardness of approximation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hardness of approximation in simple terms?

In computer science, hardness of approximation is a field that studies the algorithmic complexity of finding near-optimal solutions to optimization problems. Scope Hardness of approximation complements the study of approximation algorithms by proving, for certain problems, a limit on the factors wi…

Why does Hardness of approximation matter?

Because it connects several computer 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 Hardness of approximation?

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 Hardness of approximation.