Assignment problem

About: Assignment problem is a research topic. Over the lifetime, 7588 publications have been published within this topic receiving 172820 citations. The topic is also known as: marriage problem.

Using a genetic algorithm approach to solve the dynamic channel-assignment problem

Abstract: The Channel Assignment Problem is an NP-complete problem to assign a minimum number of channels under certain constraints to requested calls in a cellular radio system. Examples of the many approaches to solve this problem include using neural-networks, simulated annealing, graph colouring, genetic algorithms, and heuristic searches. We present a new heuristic algorithm that consists of three stages: 1) determine-lower-bound cell regular interval assignment; 2) greedy region assignment; and 3) genetic algorithm assignment. Through simulation, we show that our heuristic algorithm achieves lower bound solutions for 11 of the 13 instances of the well known Philadelphia benchmark problem. Our algorithm also has the advantage of being able to find optimum solutions faster than existing approaches that use neural networks.

How to sell hyperedges: the hypermatching assignment problem

Abstract: We are given a set of clients with budget constraints and a set of indivisible items. Each client is willing to buy one or more bundles of (at most) k items each (bundles can be seen as hyperedges in a k-hypergraph). If client i gets a bundle e, she pays bi,e and yields a net profit wi,e. The Hypermatching Assignment Problem (HAP) is to assign a set of pairwise disjoint bundles to clients so as to maximize the total profit while respecting the budgets. This problem has various applications in production planning and budget-constrained auctions and generalizes well-studied problems in combinatorial optimization: for example the weighted (unweighted) k-hypergraph matching problem is the special case of HAP with one client having unbounded budget and general (unit) profits; the Generalized Assignment Problem (GAP) is the special case of HAP with k = 1.Let e > 0 denote an arbitrarily small constant. In this paper we obtain the following main results:• We give a randomized (k + 1 + e) approximation algorithm for HAP, which is based on rounding the 1-round Lasserre strengthening of a novel LP. This is one of a few approximation results based on Lasserre hierarchies and our approach might be of independent interest. We remark that for weighted k-hypergraph matching no LP nor SDP relaxation is known to have integrality gap better than k − 1 + 1/k for general k [Chan and Lau, SODA'10].• For the relevant special case that one wants to maximize the total revenue (i.e., bi,e = wi,e), we present a local search based (k + O (√k))/2 approximation algorithm for k = O(1). This almost matches the best known (k + 1 + e)/2 approximation ratio by Berman [SWAT'00] for the (less general) weighted k-hypergraph matching problem.• For the unweighted k-hypergraph matching problem, we present a (k + 1 + e)/3 approximation in quasipolynomial time. This improves over the (k + 2)/3 approximation by Halldorsson [SODA'95] (also in quasipolynomial time). In particular this suggests that a 4/3 + e approximation for 3-dimensional matching might exist, whereas the currently best known polynomial-time approximation ratio is 3/2.

Integrating the grouping and layout problems in cellular manufacturing systems

Abstract: In this paper, the simultaneous solution of the machine grouping and layout problems in cellular manufacturing systems is explored. A model for the combined problem is presented. Since the model is complex to solve using traditional optimization techniques, a suboptimal procedure involving the use of a simulated annealing based algorithm is suggested. Results with a numerical example are presented. An alternative formulation based on quandratic assignment problem is also presented.

Sufficient Global Optimality Conditions for Bivalent Quadratic Optimization

Abstract: We prove a sufficient global optimality condition for the problem of minimizing a quadratic function subject to quadratic equality constraints where the variables are allowed to take values −1 and 1. We extend the condition to quadratic problems with matrix variables and orthonormality constraints, and in particular to the quadratic assignment problem.

Wavelength Assignment Problem on All-Optical Networks with k Fibres per Link

Abstract: Given a (possibly directed) network, the wavelength assignment problem is to minimize the number of wavelengths that must be assigned to communication paths so that paths sharing an edge are assigned different wavelengths. Our generalization to multigraphs with k parallel edges for each link (k fibres per link, with switches at nodes) may be of practical interest. While the wavelength assignment problem is NP-hard, even for a single fibre, and even in the case of simple network topologies such as rings and trees, the new model suggests many nice combinatorial problems, some of which we solve. For example, we show that for many network topologies, such as rings, stars, and specific trees, the number of wavelengths needed in the k-fibre model is less than 1/k fraction of the number required for a single fibre. We also study the existence and behavior of a gap between the minimum number of wavelengths and the natural lower bound of network congestion, the maximum number of communication paths sharing an edge. For optical stars (any size) while there is a 3/2 gap in the single fibre model, we show that with 2 fibres the gap is 0, and present a polynomial time algorithm that finds an optimal assignment. In contrast, we show that there is no fixed constant k such that for every ring and every set of communication paths the gap can be eliminated. A similar statement holds for trees. However, for rings, the gap can be made arbitrarily small, given enough fibres. The gap can even be eliminated, if the length of communication paths is bounded by a constant. We show the existence of anomalies: increasing the number of fibres may increase the gap.