For the traveling salesman problem in which the distances satisfy the triangle inequality, Christofides' heuristic produces a tour whose length is guaranteed to be less than ~ times the optimum tour length. We investigate the performance of appropriate modifications of this heuristic for the problem of finding a shortest Hamiltonian path. There are three variants of this problem. depending on the number of prespecified endpoints: zero, one, or two. It is not hard to see that, for the first two problems, the worst-case performance ratio of a Christofides-like heuristic is still -j. For the third case, we show that the ratio is i and that this bound is tight.traveling salesman problem; Hamiltonian cycle; Hamiltonian path; approximation algorithm; worst-case analysis.

Parallel machine scheduling problems concern the scheduling of n jobs on m machines to minimize some function of the job completion times. If preemption is not allowed, then most problems are not only NP-hard, but also very hard from a practical point of view. In this paper, we show that strong and fast linear programming lower bounds can be computed for an important class of machine scheduling problems with additive objective functions. Characteristic of these problems is that on each machine the order of the jobs in the relevant part of the schedule is obtained through some priority rule. To that end, we formulate these parallel machine scheduling problems as a set covering problem with an exponential number of binary variables, n covering constraints, and a single side constraint. We show that the linear programming relaxation can be solved e ciently by column generation because the pricing problem is solvable in pseudo-polynomial time. We display this approach on the problem of minimizing total weighted completion time on m identical machines. Our computational results show that the lower bound is singularly strong and that the outcome of the linear program is often integral. Moreover, they show that our branch-and-bound algorithm that uses the linear programming lower bound outperforms the previously best algorithm.

We investigate the computational complexity of scheduling multiprocessor tasks with prespecified processor allocations. We consider two criteria: minimizing schedule length and minimizing the sum of the task completion times. In addition, we investigate the complexity of problems when precedence constraints or release dates are involved.

A set of n jobs has to be scheduled on a single machine which can handle only one job at a time. Each job requires a given positive uninterrupted processing time and has a positive weight. The problem is to find a schedule that minimizes the sum of weighted deviations of the job completion times from a given common due date d, which is smaller than the sum of the processing times. We prove that this problem is NP-hard even if all job weights are equal. In addition, we present a pseudopolynomial algorithm that requires O(n2d) time and O(nd) space.

We consider the open shop, job shop, and ow shop scheduling problems with integral processing times. We give polynomial-time algorithms to determine if an instance has a schedule of length at most 3, and show that deciding if there is a schedule of length at most 4 is N Pcomplete. The latter result implies that, unless P = N P, there does not exist a polynomial-time approximation algorithm for any of these problems that constructs a schedule with length guaranteed to be strictly less than 5/4 times the optimal length. This work constitutes the rst nontrivial theoretical evidence that shop scheduling problems are hard to solve even approximately.

No abstract

No abstract

A set of unit·time tasks has to be processed on identical parallel processors subject to precedence constraints and unit·time communication delays; does there exist a schedule of length at most d? The problem has two variants, depending on whether the number of processors is res· trictively small or not. For the first variant the question can be answered in polynomial time for d =3 and is NP-complete for d =4. The second variant is solvable in polynomial time for d =5 and Np. complete for d=6. As a consequence, neither of the corresponding optimization problems has a polynomial approximation scheme, unless P=NP.1991 Mathematics Subject Classification: 90835.