In the Equal-Subset-Sum problem, we are given a set @math of @math integers and the problem is to decide if there exist two disjoint nonempty subsets @math , whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in @math time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give @math worst case Monte Carlo algorithm. This answers the open problem from Woeginger's inspirational survey. Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in @math time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in @math time and polynomial space.
and showed @math @cite_22 . The first nontrivial upper bound on @math was @math (for sufficiently large @math ) @cite_1 . Subsequently, proved that @math and showed @math . offered 500 dollars for proof or disproof of conjecture that @math for some constant @math .
@cite_1 @cite_22
1970957 1663749032
A motion picture film which includes a row of sprocket holes along one marginal edge portion and a longitudinally extending area carrying a photo-sensitive layer for photographing a selected scene, or series of scenes in a succession of stationary frames, the film being characterized in that one of its marginal edge portions carries a latent-image numerical identification of each of the successive frames in binary code form adjacent each said frame. A post-production film-making system is also described with means for transferring the binary numbers from the film to a magnetic medium and means for reading and visually displaying in arabic form the numbers on the film and the magnetic medium. Publisher Summary This chapter presents a survey of problems in combinatorial number theory. The chapter discusses problems connected with Van der Waerden's and Szemeredi's theorem. Thus the chapter gives details only if there is some important new development. According to Alfred Brauer, Schur conjectured more than 50 years ago that divided the integers into two classes at least one of them contains arbitrarily long arithmetic progressions.