Equal-Subset-Sum Faster Than the Meet-in-the-Middle

In the Equal-Subset-Sum problem, we are given a set S of n integers and the problem is to decide if there exist two disjoint nonempty subsets A,B subseteq S, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in O^*(3^(n/2)) <= O^*(1.7321^n) time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give O^*(1.7088^n) worst case Monte Carlo algorithm. This answers a question suggested by Woeginger in his inspirational survey. Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in O^*(3^n) time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in O^*(2.6817^n) time and polynomial space

Improving local search heuristics for some scheduling problems - I

Local search techniques like simulated annealing and tabu search are based on a neighborhood structure defined on a set of feasible solutions of a discrete optimization problem. For the scheduling problems $P2||C_{max}, 1|prec|\sum C_i$ and $1||\sum T_i$ we replace a simple neighborhood by a neighborhood on the set of all locally optimal solutions. This allows local search on the set of solutions that are locally optimal

Hybrid Rounding Techniques for Knapsack Problems

We address the classical knapsack problem and a variant in which an upper bound is imposed on the number of items that can be selected. We show that appropriate combinations of rounding techniques yield novel and powerful ways of rounding. As an application of these techniques, we present a linear-storage Polynomial Time Approximation Scheme (PTAS) and a Fully Polynomial Time Approximation Scheme (FPTAS) that compute an approximate solution, of any fixed accuracy, in linear time. This linear complexity bound gives a substantial improvement of the best previously known polynomial bounds.Comment: 19 LaTeX page

Scheduling identical parallel machines to minimize total weighted completion time

AbstractA branch and bound algorithm is proposed for the problem of scheduling jobs on identical parallel machines to minimize the total weighted completion time. Based upon a formulation which partitions the period of processing into unit time intervals, the lower bounding scheme is derived by performing a Lagrangean relaxation of the machine capacity constraints. A special feature is that the multipliers are obtained by a simple heuristic method which allows each lower bound to be computed in polynomial time. This bounding scheme, along with a new dominance rule, is incorporated into a branch and bound algorithm. Computational experience indicates that it is superior to known algorithms