In theoretical computer science, the term isolation lemma (or isolating lemma) refers to randomized algorithms that reduce the number of solutions to a problem to one, should a solution exist.
This is achieved by constructing random constraints such that, with non-negligible probability, exactly one solution satisfies these additional constraints if the solution space is not empty.
Isolation lemmas have important applications in computer science, such as the Valiant–Vazirani theorem and Toda's theorem in computational complexity theory.
The first isolation lemma was introduced by Valiant & Vazirani (1986), albeit not under that name.
Their isolation lemma chooses a random number of random hyperplanes, and has the property that, with non-negligible probability, the intersection of any fixed non-empty solution space with the chosen hyperplanes contains exactly one element. This suffices to show the Valiant–Vazirani theorem:
there exists a randomized polynomial-time reduction from the satisfiability problem for Boolean formulas to the problem of detecting whether a Boolean formula has a unique solution.
Mulmuley, Vazirani & Vazirani (1987) introduced an isolation lemma of a slightly different kind:
Here every coordinate of the solution space gets assigned a random weight in a certain range of integers, and the property is that, with non-negligible probability, there is exactly one element in the solution space that has minimum weight. This can be used to obtain a randomized parallel algorithm for the maximum matching problem.
Stronger isolation lemmas have been introduced in the literature to fit different needs in various settings.
For example, the isolation lemma of Chari, Rohatgi & Srinivasan (1993) has similar guarantees as that of Mulmuley et al., but it uses fewer random bits.
In the context of the exponential time hypothesis, Calabro et al. (2008) prove an isolation lemma for k-CNF formulas.
Noam Ta-Shma[1] gives an isolation lemma with slightly stronger parameters, and gives non-trivial results even when the size of the weight domain is smaller than the number of variables.
^Noam Ta-Shma (2015); A simple proof of the Isolation Lemma, in eccc
In theoretical computer science, the term isolationlemma (or isolating lemma) refers to randomized algorithms that reduce the number of solutions to...
Real-root isolationIsolationlemma, a technique used to reduce the number of solutions to a computational problem. Electrical or galvanic isolation, isolating...
some key contributions to computational complexity theory, e.g., the isolationlemma, the Valiant-Vazirani theorem, and the equivalence between random generation...
converted to finding minors in the adjacency matrix of a graph. Using the isolationlemma, a minimum weight perfect matching in a graph can be found with probability...
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proofs of the uniqueness of prime factorizations are based on Euclid's lemma: If p {\displaystyle p} is a prime number and p {\displaystyle p} divides...
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which the above result applies). In the case of convexification, Zhang's lemma majors the excess risk of the original problem using the excess risk of...
his heliocentric models. Copernicus used what is now known as the Urdi lemma and the Tusi couple in the same planetary models as found in Arabic sources...
required for safety, with actions required in each case. The Neyman–Pearson lemma of hypothesis testing says that a good criterion for the selection of hypotheses...
phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables...
syllable ending in an unreduced short vowel, this is avoided. Thus the word lemma should be divided /ˈlɛm.ə/ and not */ˈlɛ.mə/, even though the latter division...
mathematical physics from 1871 until his death in 1903. Working in relative isolation, he became the earliest theoretical scientist in the United States to...