I. Kh. Sigal

Speedup estimates for some variants of the parallel implementations of the branch-and-bound method

The efficiency of parallel implementations of the branch-and-bound method in discrete optimization problems is considered. A theoretical analysis and comparison of two parallel implementations of this method is performed. A mathematical model of the computation process is constructed and used to obtain estimates of the maximum possible speedup. Examples of problems in which none of these two parallel implementations can speed up the computations are considered.

Exact and greedy solutions of the knapsack problem: the ratio of values of objective functions

Ratios δ of the values of objective functions of optimal Boolean (or integer) to the values of greedy solutions for the knapsack problem are considered. The relationship of the parameter δ with the ratio Δ of the values of objective functions for the optimal solution of linear relaxation to the values of optimal integer solution was found. Two-sided estimates for δ and Δ were obtained. A computational experiment was conducted to investigate the ratio of δ of problems of one- and two-dimensional knapsack problems with Boolean variables. A hypothesis on asymptotic behavior of the ratio δ with growth of the number of problem variables was formulated.

Ratios of optimal values of objective functions of the knapsack problem and its linear relaxation

The ratios of the values of objective functions for optimal solutions of linear and integer knapsack problems are considered. Estimates for these ratios are obtained. One-dimensional and multi-dimensional knapsack problems with Boolean variables are studied experimentally. For these problems, a hypothesis is formulated on the asymptotic behavior of the ratio as the number of variables grows.

On a lower bound on the computational complexity of a parallel implementation of the branch-and-bound method

We study parallel complexity of the branch-and-bound method for optimization problems. We consider a standard implementation scheme for the branch-and-bound method on a parallel system, in which first only one processor is working, and then the resulting subtasks are given out to other processors. For this scheme, we give a lower bound on the parallel complexity independent of the problem. We study the complexity of this scheme for the Boolean knapsack problem. For a classical algorithmically hard example, we obtain parallel complexity bounds and show that these bounds coincide in order with each other and with the common lower bound on parallel complexity. Thus, we show that the common lower bound is achieved, in the order, for some optimization problems.

A combined parallel algorithm for solving the knapsack problem

Parallel implementations of a combined branch-and-bound algorithm for the knapsack problem with one constraint are considered. By the combined algorithm we mean an algorithm in which two methods of branching are implemented, the method based on an estimate of the upper bound and the method of one-sided branching based on the vector. An approach combining parallel implementations of the brunch-and-bound method and the heuristic search is proposed and implemented.

Accounting for the time characteristics of a class of scheduling problems for moving processor

Consideration was given to scheduling service of the stationary objects distributed over a one-dimensional zone by a processor moving within this zone. Servicing is performed in the course of two processor passes, direct, from the initial to the final point of the zone, and reverse. Schedules are designed with account for the early start of service prescribed to the objects and/or the deadlines of service completion. For the problems accounting for the early timing of service start, studied were the questions of finding the criterion-optimal time schedules. For the problems with prescribed deadlines, existence of schedules meeting such deadlines was investigated.

Application of parallel heuristic algorithms for speeding up parallel implementations of the branch-and-bound method

A scheme for the parallel implementation of the combined branch-and-bound method and heuristic algorithms is proposed. Results of computations for the one-dimensional Boolean knapsack problem are presented that demonstrate the efficiency of the proposed approach. The main factors that affect the speedup of the solution when local optimization is used are discussed.