Weakly hard problems

Abstract

A weak completeness phenomenon is investigated in the complexity class E=DTIME(2/sup linear/). According to standard terminology, a language H is /spl lessub msup P/-hard for E if the set P/sub m/(H), consisting of all languages A/spl lessub msup P/H, contains the entire class E. A language C is /spl lessub msup P/-complete for E if it is /spl lessub msup P/-hard for E and is also an element of E. Generalizing this, a language H is weakly /spl lessub msup P/-hard for E if the set P/sub m/(H) does not have measure 0 in E. A language C is weakly /spl lessub msup P/-complete for E if it is weakly /spl lessub msup P/-hard for E and is also an element of E. The main result of this paper is the construction of a language that is weakly /spl lessub msup P/-complete, but not /spl lessub msup P/-complete, for E. The existence of such languages implies that previously known strong lower bounds on the complexity of weakly /spl lessub msup P/-hard problems for E are indeed more general than the corresponding bounds for /spl lessub msup P/-hard problems for E. The proof of this result introduces a new diagonalization method, called martingale diagonalization. Using this method, one simultaneously develops an infinite family of polynomial time computable martingales (betting strategies) and a corresponding family of languages that defeat these martingales (i.e. prevent them from winning too much money), while also pursuing another agenda. Martingale diagonalization may be useful for a variety of applications.<>