On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

Effective parallelization strategy for the solution of subset sum problems by the branch-and-bound method

An easily implementable recursive parallelization strategy for solving the subset sum problem by the branch-and-bound method is proposed. Two different frontal and balanced variants of this strategy are compared. On an example of a particular case of the subset sum problem we show that the balanced variant is more effective than the frontal one. Moreover, we show that, for the considered particular case of the subset sum problem, the balanced variant is also time optimal.