L-reduction

Key takeaways

L-reduction 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 L-reduction to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of L-reduction from memory before moving on to harder problems.

Reference excerpt

In computer science, particularly the study of approximation algorithms, an L-reduction ("linear reduction") is a transformation of optimization problems which linearly preserves approximability features; it is one type of approximation-preserving reduction. L-reductions in studies of approximability of optimization problems play a similar role to that of polynomial reductions in the studies of computational complexity of decision problems. The term L reduction is sometimes used to refer to log-space reductions, by analogy with the complexity class L, but this is a different concept.

Definition

Before giving the definition, we recall some concepts of optimization problems, illustrated with the travelling salesman problem (TSP). First an instance is the input of the problem, i.e. the information we need to compute a solution. An instance for TSP is a finite set of cities and the distances between them. A solution is a tour that visits all the cities. In the case of TSP, the cost of a solution is the length of the tour. We write O P T T S P ( x ) {\displaystyle \mathrm {OPT_{TSP}} (x)} to be the cost of an optimal solution, i.e. the length of a shortest tour. Let A and B be optimization problems and cA and cB their respective cost functions. A pair of functions f and g is an L-reduction if all of the following conditions are met:

functions f and g are computable in polynomial time, if x is an instance of problem A, then f(x) is an instance of problem B, if y' is a solution to f(x), then g(y' ) is a solution to x, there exists a positive constant α such that

O P T B ( f ( x ) ) ≤ α O P T A ( x ) {\displaystyle \mathrm {OPT_{B}} (f(x))\leq \alpha \mathrm {OPT_{A}} (x)} , there exists a positive constant β such that for every solution y' to f(x)

| O P T A ( x ) − c A ( g ( y ′ ) ) | ≤ β | O P T B ( f ( x ) ) − c B ( y ′ ) | {\displaystyle |\mathrm {OPT_{A}} (x)-c_{A}(g(y'))|\leq \beta |\mathrm {OPT_{B}} (f(x))-c_{B}(y')|} .

Example Here is a L-reduction from MAX 3-SAT to MAX 2-SAT. Consider a MAX 3-SAT instance φ := ⋀ i = 1 m ( ℓ i 1 ∨ ℓ i 2 ∨ ℓ i 3 ) {\displaystyle \varphi :=\bigwedge _{i=1}^{m}(\ell _{i}^{1}\lor \ell _{i}^{2}\lor \ell _{i}^{3})} where ℓ i 1 {\displaystyle \ell _{i}^{1}} , ℓ i 2 {\displaystyle \ell _{i}^{2}} and ℓ i 3 {\displaystyle \ell _{i}^{3}} are literals. The function f {\displaystyle f} of our L-reduction is defined as follows. The MAX 2-SAT instance f ( φ ) {\displaystyle f(\varphi )} is the formula:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with L-reduction

Start with the simplest possible case. Write down what L-reduction 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 L-reduction 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 L-reduction 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 L-reduction

In research: L-reduction 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 L-reduction 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: L-reduction is common in secondary-school and first-year university syllabi. It links to neighbouring topics Approximation algorithms, Reduction (complexity), Theoretical computer science stubs, so understanding it makes those chapters shorter.
In everyday life: Look for L-reduction 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 L-reduction in 20 minutes

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

Frequently asked questions

What is L-reduction in simple terms?

In computer science, particularly the study of approximation algorithms, an L-reduction ("linear reduction") is a transformation of optimization problems which linearly preserves approximability features; it is one type of approximation-preserving reduction. L-reductions in studies of approximabili…

Why does L-reduction 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 L-reduction?

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 L-reduction.