Math 2.0

Computer science Randomized, circuit, counting and interactive complexity Conjecture

Exponential time hypothesis

Builds on Boolean satisfiability problemBig O notation

Statement

Let k≥3k \ge 3 be an integer. The problem kk-SAT asks whether a Boolean formula in conjunctive normal form with at most kk literals per clause is satisfiable. Let nn be the number of variables of the formula. Let sks_k be the infimum of the real numbers δ\delta such that kk-SAT can be solved by an algorithm in time O(2δn)O(2^{\delta n}).

Hypothesis. s3>0.s_3 > 0. That is, there is a constant c>0c > 0 such that no algorithm solves 3-SAT in time O(2cn)O(2^{c n}).

Remarks

The hypothesis is open, and it implies P≠NP\mathrm{P} \ne \mathrm{NP}. Here time is measured in big O notation in the number of variables, which is a natural measure since brute force takes about 2n2^n steps. The best known algorithms for 3-SAT run in time O(cn)O(c^n) for constants cc somewhat below 1.331.33, which is consistent with the hypothesis.

Impagliazzo, Paturi and Zane proved the sparsification lemma. With it, the hypothesis is equivalent to the same statement with nn replaced by the number of clauses. It then implies that many NP-complete problems, such as finding a kk-clique or a vertex cover, need time 2Ω(n)2^{\Omega(n)} or nΩ(k)n^{\Omega(k)} in suitable forms.

The strong exponential time hypothesis says that sk→1s_k \to 1 as k→∞k \to \infty. It implies the exponential time hypothesis.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Big O notation, concept, no evidence levelBig O notationconceptConjunctive normal form, concept, no evidence levelConjunctive normal fo…conceptNP, concept, no evidence levelNPconceptP, concept, no evidence levelPconceptProposition, concept, no evidence levelPropositionconceptSatisfiability, concept, no evidence levelSatisfiabilityconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionPropositional logic, concept, no evidence levelPropositional logicconceptBoolean satisfiability problem, problem, no evidence levelBoolean satisfiabilit…problemExponential time hypothesis, conjecture, no evidence levelExponential time hypo…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Big O notation concept
  2. Conjunctive normal form concept
  3. NP concept
  4. P concept
  5. Proposition concept
  6. Satisfiability concept
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Negation definition
  11. Biconditional definition
  12. Propositional logic concept
  13. Boolean satisfiability problem problem
  14. Exponential time hypothesis conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Exponential_time_hypothesis
About this page
Tag5ZRQC3DG7W
URIm2://5ZRQC3DG7W
Kindconjecture
WritingW0 stub
MSC68Q17, 68Q25, 68Q27
StatusThis copy of the site has no build data. Run m2 status 5ZRQC3DG7W.
AboutBoolean satisfiability problem
UsesBoolean satisfiability problem, Big O notation
Changed3 November 2026

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