Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, February 24, 2011, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Dominik Scheder

Mahaney's Theorem: A Complexity Theory Classic

For a language L over {0,1}^*, its count function c(n) is the number of words in L of size at most n. For example, if L is the language of palindromes, then c(n) is approximately 2^(n/2). A language is sparse if its count function c(n) is bounded by some polynomial p(n).

There are sparse languages that are undecidable, and there are decidable sparse languages with arbitrarily high time complexity. However, Mahaney's Theorem states that if there is an NP-complete sparse language L, then P=NP. The proof is constructive, i.e., gives an algorithm quickly solving SAT. The challenge is that the algorithm cannot make any membership queries to L, since, as said, L might even be undecidable.

Mahaney's Theorem is a classic, and may be older than many people in the audience (it dates back to 1982). Its proof is fun and nicely highlights the weird ways in which you often have to think in complexity theory.

References:
Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis by Steve Mahaney, JCSS 1982.

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