257 research outputs found
Breaking symmetries to rescue Sum of Squares in the case of makespan scheduling
The Sum of Squares (\sos{}) hierarchy gives an automatized technique to
create a family of increasingly tight convex relaxations for binary programs.
There are several problems for which a constant number of rounds of this
hierarchy give integrality gaps matching the best known approximation
algorithms. For many other problems, however, ad-hoc techniques give better
approximation ratios than \sos{} in the worst case, as shown by corresponding
lower bound instances. Notably, in many cases these instances are invariant
under the action of a large permutation group. This yields the question how
symmetries in a formulation degrade the performance of the relaxation obtained
by the \sos{} hierarchy. In this paper, we study this for the case of the
minimum makespan problem on identical machines. Our first result is to show
that rounds of \sos{} applied over the \emph{configuration linear
program} yields an integrality gap of at least , where is the
number of jobs. Our result is based on tools from representation theory of
symmetric groups. Then, we consider the weaker \emph{assignment linear program}
and add a well chosen set of symmetry breaking inequalities that removes a
subset of the machine permutation symmetries. We show that applying
rounds of the SA hierarchy to this stronger
linear program reduces the integrality gap to , which yields a
linear programming based polynomial time approximation scheme. Our results
suggest that for this classical problem, symmetries were the main barrier
preventing the \sos{}/ SA hierarchies to give relaxations of polynomial
complexity with an integrality gap of~. We leave as an open
question whether this phenomenon occurs for other symmetric problems
Lift-and-Round to Improve Weighted Completion Time on Unrelated Machines
We consider the problem of scheduling jobs on unrelated machines so as to
minimize the sum of weighted completion times. Our main result is a
-approximation algorithm for some fixed , improving upon the
long-standing bound of 3/2 (independently due to Skutella, Journal of the ACM,
2001, and Sethuraman & Squillante, SODA, 1999). To do this, we first introduce
a new lift-and-project based SDP relaxation for the problem. This is necessary
as the previous convex programming relaxations have an integrality gap of
. Second, we give a new general bipartite-rounding procedure that produces
an assignment with certain strong negative correlation properties.Comment: 21 pages, 4 figure
A hybrid shifting bottleneck-tabu search heuristic for the job shop total weighted tardiness problem
In this paper, we study the job shop scheduling problem with the objective of minimizing the total weighted tardiness. We propose a hybrid shifting bottleneck - tabu search (SB-TS) algorithm by replacing the reoptimization step in the shifting bottleneck (SB) algorithm by a tabu search (TS). In terms of the shifting bottleneck heuristic, the proposed tabu search optimizes the total weighted tardiness for partial schedules in which some machines are currently assumed to have infinite capacity. In the context of tabu search, the shifting bottleneck heuristic features a long-term memory which helps to diversify the local search. We exploit this synergy to develop a state-of-the-art algorithm for the job shop total weighted tardiness problem (JS-TWT). The computational
effectiveness of the algorithm is demonstrated on standard benchmark instances from the literature
Modeling Industrial Lot Sizing Problems: A Review
In this paper we give an overview of recent developments in the field of modeling single-level dynamic lot sizing problems. The focus of this paper is on the modeling various industrial extensions and not on the solution approaches. The timeliness of such a review stems from the growing industry need to solve more realistic and comprehensive production planning problems. First, several different basic lot sizing problems are defined. Many extensions of these problems have been proposed and the research basically expands in two opposite directions. The first line of research focuses on modeling the operational aspects in more detail. The discussion is organized around five aspects: the set ups, the characteristics of the production process, the inventory, demand side and rolling horizon. The second direction is towards more tactical and strategic models in which the lot sizing problem is a core substructure, such as integrated production-distribution planning or supplier selection. Recent advances in both directions are discussed. Finally, we give some concluding remarks and point out interesting areas for future research
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