Isolation lemma


In theoretical computer science, the term isolation 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, 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.
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 has similar guarantees as that of Mulmuley et al., but it uses fewer random bits.
In the context of the exponential time hypothesis, prove an isolation lemma for k-CNF formulas.
Noam Ta-Shma 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.

The isolation lemma of Mulmuley, Vazirani, and Vazirani

It is remarkable that the lemma assumes nothing about the nature of the family : for instance may include all nonempty subsets. Since the weight of each set in is between and on average there will be sets of each possible weight.
Still, with high probability, there is a unique set that has minimum weight.



Suppose we have fixed the weights of all elements except an element x. Then x has a threshold weight α, such that if the weight w of x is greater than α, then it is not contained in any minimum-weight subset, and if, then it is contained in some sets of minimum weight. Further, observe that if, then every minimum-weight subset must contain x. Thus, ambiguity about whether a minimum-weight subset contains x or not can happen only when the weight of x is exactly equal to its threshold; in this case we will call x "singular". Now, as the threshold of x was defined only in terms of the weights of the other elements, it is independent of w, and therefore, as w is chosen uniformly from,
and the probability that some x is singular is at most n/N. As there is a unique minimum-weight subset iff no element is singular, the lemma follows.
Remark: The lemma holds with since it is possible that some x has no threshold value.





This is a restatement version of the above proof, due to Joel Spencer.
For any element x in the set, define
Observe that depends only on the weights of elements other than x, and not on w itself. So whatever the value of, as w is chosen uniformly from, the probability that it is equal to is at most 1/N. Thus the probability that for some x is at most n/N.
Now if there are two sets A and B in with minimum weight, then, taking any x in A\B, we have
and as we have seen, this event happens with probability at most n/N.

Examples/applications