In this paper, we investigate the parametric weight knapsack problem, in which the item weights are affine functions of the formwi(λ) = ai + λ ·bi for i ∈ {1, . . . ,n} depending on a real-valued parameter λ. The aim is to provide a solution for all values of the parameter. It is well-known that any exact algorithm for the problem may need to output an exponential number of knapsack solutions. ...

Pruhs and Woeginger [10] prove the existence of FPTAS’s for a general class of minimization and maximization subset selection problems. Without losing generality from the original framework, we prove how better asymptotic worst-case running times can be achieved if a ρ-approximation algorithm is available, and in particular we obtain matching running times between maximization and minimization ...

We study the convergence time of Nash dynamics in two classes of congestion games – constant player congestion games and bounded jump congestion games. It was shown by Ackermann and Skopalik [2] that even 3-player congestion games are PLS-complete. We design an FPTAS for congestion games with constant number of players. In particular, for any > 0, we establish a stronger result, namely, any seq...

A fully polynomial time approximation scheme (FPTAS) with run time O( n m εm−1 ) is developed for a problem which combines common due window assignment and scheduling n jobs on m identical parallel machines. The problem criterion is bottleneck (min-max) such that the maximum cost, which includes job earliness, job tardiness and due window size costs, is minimized.

We study the classical 0-1 knapsack problem with additional restrictions on pairs of items. A con ict constraint states that from a certain pair of items at most one item can be contained in a feasible solution. Reversing this condition, we obtain a forcing constraint stating that at least one of the two items must be included in the knapsack. A natural way for representing these constraints is...

We give a reduction from clique to establish that sparse PCA is NP-hard. The reduction has a gap which we use to exclude an FPTAS for sparse PCA (unless P=NP). Under weaker complexity assumptions, we also exclude polynomial constant-factor approximation algorithms.

We consider the scheduling problem of minimizing the makespan on a single machine with step-improving job processing times around a common critical date. For this problem we give an NP-hardness proof, a fast pseudo-polynomial time algorithm, an FPTAS, and an on-line algorithm with best possible competitive ratio.

We discuss three FPTASes for the Multi-Objective Shortest Path problem. Two of which are known in the literature, the third is a newly proposed FPTAS, aimed at exploiting small Pareto curves. The FPTASes are analyzed, empirically, based on worst case complexity, average case complexity and smoothed complexity. We also analyze the size of the Pareto curve under different conditions.