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Planar SAT

Planar SAT is a mathematics 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 Planar SAT rather than just read about it. In short: In computer science, the planar 3-satisfiability problem (abbreviated PLANAR 3SAT or PL3SAT) is an extension of the classical Boolean 3-satisfiability problem to a planar incidence graph. In other words, it asks whether the variables of a given Boolean formula—whose incidence graph consisting of variables and clauses can be embedded on a plane—can be consistently replaced by the values TRUE or FALSE in such a way th…

Planar SAT — main illustration
Planar SAT — illustration

Key takeaways

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

Reference excerpt

In computer science, the planar 3-satisfiability problem (abbreviated PLANAR 3SAT or PL3SAT) is an extension of the classical Boolean 3-satisfiability problem to a planar incidence graph. In other words, it asks whether the variables of a given Boolean formula—whose incidence graph consisting of variables and clauses can be embedded on a plane—can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable. Like 3SAT, PLANAR-SAT is NP-complete, and is commonly used in reductions.

Definition Every 3SAT problem can be converted to an incidence graph in the following manner: For every variable v i {\displaystyle v_{i}} , the graph has one corresponding node v i {\displaystyle v_{i}} , and for every clause c j {\displaystyle c_{j}} , the graph has one corresponding node c j . {\displaystyle c_{j}.} An edge ( v i , c j ) {\displaystyle (v_{i},c_{j})} is created between variable v i {\displaystyle v_{i}} and clause c j {\displaystyle c_{j}} whenever v i {\displaystyle v_{i}} or ¬ v i {\displaystyle \lnot v_{i}} is in c j {\displaystyle c_{j}} . Positive and negative literals are distinguished using edge colorings. The formula is satisfiable if and only if there is a way to assign TRUE or FALSE to each variable node such that every clause node is connected to at least one TRUE by a positive edge or FALSE by a negative edge. A planar graph is a graph that can be drawn on the plane in a way such that no two of its edges cross each other. Planar 3SAT is a subset of 3SAT in which the incidence graph of the variables and clauses of a Boolean formula is planar. It is important because it is a restricted variant, and is still NP-complete. Many problems (for example games and puzzles) cannot represent non-planar graphs. Hence, Planar 3SAT provides a way to prove those problems to be NP-hard.

Proof of NP-completeness

The following proof sketch follows the proof of D. Lichtenstein. Trivially, PLANAR 3SAT is in NP. It is thus sufficient to show that it is NP-hard via reduction from 3SAT. This proof makes use of the fact that ( ¬ a ∨ ¬ b ∨ c ) ∧ ( a ∨ ¬ c ) ∧ ( b ∨ ¬ c ) {\displaystyle (\lnot a\lor \lnot b\lor c)\land (a\lor \lnot c)\land (b\lor \lnot c)} is equivalent to ( a ∧ b ) ↔ c {\displaystyle (a\land b)\leftrightarrow c} and that ( a ∨ ¬ b ) ∧ ( ¬ a ∨ b ) {\displaystyle (a\lor \lnot b)\land (\lnot a\lor b)} is equivalent to a ↔ b {\displaystyle a\leftrightarrow b} . First, draw the incidence graph of the 3SAT formula. Since no two variables or clauses are connected, the resulting graph will be bipartite. Suppose the resulting graph is not planar. For every crossing of edges (a, c1) and (b, c2), introduce nine new variables a1, b1, α, β, γ, δ, ξ, a2, b2, and replace every crossing of edges with a crossover gadget shown in the diagram. It consists of the following new clauses:

… excerpt ends here. Continue reading the full article.

Illustrations

Planar SAT: Example of a planar SAT problem. The black edges correspond to non-inverted variables and the red edges correspond to inverted variables.
Example of a planar SAT problem. The black edges correspond to non-inverted variables and the red edges correspond to inverted variables.
Planar SAT: Left side is a crossing; right side is the crossover gadget. The small dots represent clauses. The black and red edges correspond to non-inverted and inverted variables respectively.
Left side is a crossing; right side is the crossover gadget. The small dots represent clauses. The black and red edges correspond to non-inverted and inverted variables respectively.

Worked examples

Example 1 — a first encounter with Planar SAT

Start with the simplest possible case. Write down what Planar SAT claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Planar SAT 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 Planar SAT 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 Planar SAT

In research
Planar SAT appears in mathematics 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 Planar SAT 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
Planar SAT is common in secondary-school and first-year university syllabi. It links to neighbouring topics Boolean algebra, Electronic design automation, NP-complete problems, so understanding it makes those chapters shorter.
In everyday life
Look for Planar SAT 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 Planar SAT in 20 minutes

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

Frequently asked questions

What is Planar SAT in simple terms?

In computer science, the planar 3-satisfiability problem (abbreviated PLANAR 3SAT or PL3SAT) is an extension of the classical Boolean 3-satisfiability problem to a planar incidence graph. In other words, it asks whether the variables of a given Boolean formula—whose incidence graph consisting of va…

Why does Planar SAT matter?

Because it connects several mathematics 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 Planar SAT?

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 Planar SAT.

Tags

  • Boolean algebra
  • Electronic design automation
  • NP-complete problems
  • Satisfiability problems

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