Hans Kellerer

Semi on-line algorithms for the partition problem

The partition problem is one of the basic NP-complete problems. While an efficient heuristic for the optimization version, which is equivalent to minimizing the makespan on two identical machines, is known with worst-case ratio 1211, no deterministic heuristic for the on-line problem can have a worst-case ratio lower than 32. In this paper we investigate three different semi on-line versions of the partition problem. In the first case, we assume that a buffer of length k is available to maintain k items. In the second case, two parallel processors are available which assign each item independently to the partition sets. The best of the two produced solutions is chosen. Finally, in the third problem the total sum of the items is known in advance. For each version we propose a heuristic and investigate its worst-case ratio. All algorithms have a worst-case ratio of 43 which is shown to be the best possible worst-case ratio.

Approximation algorithms for knapsack problems with cardinality constraints

Abstract We address a variant of the classical knapsack problem in which an upper bound is imposed on the number of items that can be selected. This problem arises in the solution of real-life cutting stock problems by column generation, and may be used to separate cover inequalities with small support within cutting-plane approaches to integer linear programs. We focus our attention on approximation algorithms for the problem, describing a linear-storage Polynomial Time Approximation Scheme (PTAS) and a dynamic-programming based Fully Polynomial Time Approximation Scheme (FPTAS). The main ideas contained in our PTAS are used to derive PTASs for the knapsack problem and its multi-dimensional generalization which improve on the previously proposed PTASs. We finally illustrate better PTASs and FPTASs for the subset sum case of the problem in which profits and weights coincide.

Selecting Portfolios with Fixed Costs and Minimum Transaction Lots

The original Markowitz model of portfolio selection has received a widespread theoretical acceptance and it has been the basis for various portfolio selection techniques. Nevertheless, this normative model has found relatively little application in practice when some additional features, such as fixed costs and minimum transaction lots, are relevant in the portfolio selection problem. In this paper different mixed-integer linear programming models dealing with fixed costs and possibly minimum lots are introduced. Due to the high computational complexity of the models, heuristic procedures, based on the construction and optimal solution of mixed integer subproblems, are proposed. Computational results obtained using data from the Milan Stock Exchange show how the proposed heuristics yield very good solutions in a short computational time and make possible some interesting financial conclusions on the impact of fixed costs and minimum lots on portfolio composition.

Approximability and nonapproximability results for minimizing total flow time on a single machine

We consider the problem of scheduling n jobs that are released over time on a single machine in order to minimize the total ??flow time??. This problem is well-??known to be NP??-complete, and the best polynomial time approximation algorithms constructed so far had (more or less trivial)?? worst-??case performance guarantees of O??(n).???? In this paper, we present one positive and one negative result on polynomial time approximations for the minimum total ??flow time problem??. The positive result is the first approxima??tion algorithm with a sublinear worst-??case performance guarantee of O(\sqrt{n}). This algorithm is based on resolving the preemptions of the corresponding optimum preemptive schedule??. The performance guarantee of our approximation algorithm is not far from best possible as our second, negative result demonstrates.?? Unless P=NP, no polynomial time approxima??tion algorithm for minimum total ??flow time can have a worst-??case performance guarantee of O(n^{1/2 - \epsilon}) for any \epsilon > 0. ?? ?? ???? Keywords:?? scheduling, approximation algorithm, worst-??case analysis, total flow time, release time, single machine??.

Fully Polynomial Approximation Schemes for a Symmetric Quadratic Knapsack Problem and its Scheduling Applications

We design a fully polynomial-time approximation scheme (FPTAS) for a knapsack problem to minimize a symmetric quadratic function. We demonstrate how the designed FPTAS can be adopted for several single machine scheduling problems to minimize the sum of the weighted completion times. The applications presented in this paper include problems with a single machine non-availability interval (for both the non-resumable and the resumable scenarios) and a problem of planning a single machine maintenance period; the latter problem is closely related to a single machine scheduling problem with two competing agents. The running time of each presented FPTAS is strongly polynomial.

The Multiple Subset Sum Problem

In the {\em multiple subset sum problem} (MSSP) items from a given ground set are selected and packed into a given number of identical bins such that the sum of the item weights in every bin does not exceed the bin capacity and the total sum of the weights of the items packed is as large as possible. This problem is a relevant special case of the multiple knapsack problem, for which the existence of a polynomial-time approximation scheme (PTAS) is an important open question in the field of knapsack problems. One main result of the present paper is the construction of a PTAS for MSSP. For the bottleneck case of the problem, where the minimum total weight contained in any bin is to be maximized, we describe a 2/3-approximation algorithm and show that this is the best possible approximation ratio. Moreover, PTASs are derived for the special cases in which either the number of bins or the number of different item weights is constant. We finally show that, even for the case of only two bins, no fully PTAS exists for both versions of the problem.

The exact LPT-bound for maximizing the minimum completion time

We consider the problem of assigning a set of jobs to a system of m identical processors in order to maximize the earliest processor completion time. It was known that the LPT-heuristic gives an approximation of worst case ratio at most 34. In this note we show that the exact worst case ratio of LPT is (3m - 1)/(4m - 2).

Improved Dynamic Programming in Connection with an FPTAS for the Knapsack Problem

A vector merging problem is introduced where two vectors of length n are merged such that the k-th entry of the new vector is the minimum over ℓ of the ℓ-th entry of the first vector plus the sum of the first k − ℓ + 1 entries of the second vector. For this problem a new algorithm with O(n log n) running time is presented thus improving upon the straightforward O(n2) time bound.The vector merging problem can appear in different settings of dynamic programming. In particular, it is applied for a recent fully polynomial time approximation scheme (FPTAS) for the classical 0–1 knapsack problem by the same authors.

Scheduling problems for parallel dedicated machines under multiple resource constraints

The paper considers scheduling problems for parallel dedicated machines subject to resource constraints. A fairly complete computational complexity classification is obtained, a number of polynomial-time algorithms are designed. For the problem with a fixed number of machines in which a job uses at most one resource of unit size a polynomial-time approximation scheme is offered.

An efficient fully polynomial approximation scheme for the Subset-Sum problem

Given a set of n positive integers and a knapsack of capacity c, the Subset-Sum Problem is to find a subset the sum of which is closest to c without exceeding the value c. In this paper we present a fully polynomial approximation scheme which solves the Subset-Sum Problem with accuracy e in time O(min{n . 1/e, n + 1/e2 log(1/e)}) and space O(n + 1/e). This scheme has a better time and space complexity than previously known approximation schemes. Moreover, the scheme always finds the optimal solution if it is smaller than (1 - e)c. Computational results show that the scheme efficiently solves instances with up to 5000 items with a guaranteed relative error smaller than 1/1000.