Given an acyclic directed graph and two distinct nodes s and t, we consider the problem of finding k disjoint paths from s to t satisfying some objective. We consider four objectives, MinMax, Balanced, MinSum-MinMin, and MinSum-MinMax. We use the algorithm by Perl-Shiloach and labelling and scaling techniques to devise an FPTAS for the first three objectives. For the forth one, we propose a gen...

We consider the problem of maximizing the minimum load for machines that are controlled by selfish agents, who are only interested in maximizing their own profit. Unlike the regular load balancing problem, this problem has not been considered in this context before. For a constant number of machines, m, we show a monotone polynomial time approximation scheme (PTAS) with running time that is lin...

We consider the two-parallel machines scheduling problem, with the aim of minimizing the maximum lateness and the makespan. Formally, the problem is defined as follows. We have to schedule a set J of n jobs on two identical machines. Each job i ∈ J has a processing time pi and a delivery time qi. Each machine can only perform one job at a given time. The machines are available at time t = 0 and...

In this paper the 0-1 Multiple-Choice Knapsack Problem (0-1 MCKP), a generalization of the classical 0-1 Knapsack problem, is addressed. We present a fast Fully Polynomial Time Approximation Scheme (FPTAS) for the 0-1 MCKP, which yields a better time bound than known algorithms. In particular it produces a (1+ ) approximate solution and runs in O(nm/ ) time, where n is the number of items and m...

We study the problems of scheduling a set of nonpreemptive jobs on a single machine and identical parallel machines, where the processing time of a job is a piecewise linear nonincreasing function of its start time. The objective is to minimize makespan. We first give a fully polynomialtime approximation scheme (FPTAS) for the case with a single machine. We then generalize the result to the cas...

We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixedinteger sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary pol...

In TSP with profits we have to find an optimal tour and a set of customers satisfying a specific constraint. In this paper we analyze three different variants of TSP with profits that are NP-hard in general. We study their computational complexity on special structures of the underlying graph, both in the case with and without service times to the customers. We present polynomial algorithms for...

In this paper we study an extension of the single machine scheduling problem with the total weighted completion time objective, where there is a single non-renewable resource consumed by the jobs, having an initial stock and some additional replenishments over time. We prove that this problem is NP-hard in the strong sense, and provide an FPTAS for a special case with two supply dates.