Petra Schuurman

Polynomial time approximation algorithms for machine scheduling: ten open problems

We discuss what we consider to be the 10 most vexing open questions in the area of polynomial time approximation algorithms for NP-hard deterministic machine scheduling problems. We summarize what is known on these problems, we discuss related results, and we provide pointers to the literature. Copyright

Non-Approximability Results for Scheduling Problems with Minsum Criteria

We provide several non-approximability results for deterministic scheduling problems whose objective is to minimize the total job completion time. Unless , none of the problems under consideration can be approximated in polynomial time within arbitrarily good precision. Most of our results are derived by APX-hardness proofs.We show that, whereas scheduling on unrelated machines with unit weights is polynomially solvable, the problem becomes APX-hard if release dates or weights are added. We further show APX-hardness for scheduling in flow shops, job shops, and open shops. We also investigate the problems of scheduling on parallel machines with precedence constraints and unit processing times, and two variants of the latter problem with unit communication delays; for these problems we provide lower bounds on the worst-case behavior of any polynomial-time approximation algorithm through the gap-reduction technique.

Non-approximability Results for Scheduling Problems with Minsum Criteria

We provide several non-approximability results for deterministic scheduling problems whose objective is to minimize the total job completion time. Unless P = NP, none of the problems under consideration can be approximated in polynomial time within arbitrarily good precision. Most of our results are derived by Max SNP hardness proofs. Among the investigated problems are: scheduling unrelated machines with some additional features like job release dates, deadlines and weights, scheduling flow shops, and scheduling open shops.

A polynomial time approximation scheme for the two-stage multiprocessor flow shop problem

In this paper we investigate the two-stage multiprocessor flow shop scheduling problem F2(P)|·|Cmax, where the numbers m1 and m2 of machines available in the two stages are part of the input. We demonstrate the existence of a polynomial time approximation scheme for this problem. This result solves the simplest case of an open problem that has been posed by Leslie Hall in a recent paper (Hall, 1995). An extension of our algorithm yields an approximation scheme for the closely related two-stage multiprocessor job shop problem.

Approximation algorithms for the multiprocessor open shop scheduling problem

We investigate the multiprocessor multistage open shop scheduling problem. In this variant of the open shop model, there are s stages, each consisting of a number of parallel identical machines. Each job consists of s operations, one for each stage, that can be executed in any order. The goal is to nd a nonpreemptive schedule that minimizes the makespan. We derive two approximation results for this NP-hard problem. First, we demonstrate the existence of a polynomial time approximation algorithm with worst case ratio 2 for the case that the number s of stages is part of the input. This algorithm is based on Racsmanys concept of dense schedules. Secondly, for the multiprocessor two stage open shop problem we derive a family of polynomialtime approximation algorithms whose worst case ratios can be made arbitrarily close to 3/2.

Recognizing DNA graphs is difficult

DNA graphs are the vertex induced subgraphs of De Bruijn graphs over a four letter alphabet. In this paper, we prove the NP-hardness of various recognition problems for subgraphs of De Bruijn graphs; in particular, the recognition of DNA graphs is shown to be NP-hard. As a consequence, two open questions from a recent paper by Biazewicz et al. (Discrete Appl. Math. 98, (1999) 1) are answered in the negative.

A table of state complexity bounds for binary linear codes

This article contains a table of bounds on the state complexity of binary linear codes with length smaller than 25. General results on the state complexity of binary linear codes with low dimension or low minimum distance are included.

DNA Sequencing, Eulerian Graphs, and the Exact Perfect Matching Problem

We investigate the computational complexity of a combinatorial problem that arises in DNA sequencing by hybridization: The input consists of an integer l together with a set S of words of length k over the four symbols A, C, G, T. The problem is to decide whether there exists a word of length l that contains every word in S at least once as a subword, and does not contain any other subword of length k.The computational complexity of this problem has been open for some time, and it remains open. What we prove is that this problem is polynomial time equivalent to the exact perfect matching problem in bipartite graphs, which is another infamous combinatorial optimization problem of unknown computational complexity.

Performance guarantees of jump neighborhoods on restricted related parallel machines

We study the performance of two popular jump neighborhoods on the classical scheduling problem of minimizing the makespan on related parallel machines under the additional restriction that jobs are only allowed to be scheduled on a subset of machines. In particular, we analyze the performance guarantee of local optima with respect to the jump and the lexicographical jump neighborhood.

A polynomial time equivalence between DNA sequencing and the exact perfect matching problem

We investigate the computational complexity of a combinatorial problem that arises in DNA sequencing by hybridization: The input consists of an integer @? together with a set S of words of length k over the four symbols A, C, G, T. The problem is to decide whether there exists a word of length @? that contains every word in S at least once as a subword, and does not contain any other subword of length k. The computational complexity of this problem has been open for some time, and it remains open. What we prove is that this problem is polynomial time equivalent to the exact perfect matching problem in bipartite graphs, which is another infamous combinatorial optimization problem of unknown computational complexity.