Approximation results for the weighted partition problem

We present several new standard and differential approximation results for the P₄-partition problem using the Hassin and Rubinstein algorithm [Information Processing Letters 63 (1997) 63–67]. Those results concern both minimization and maximization versions of the problem. However, the main point of this paper lies in the establishment of the robustness of this algorithm, in the sense that it provides good quality solutions for a variety of versions of the problem, under both standard and differential approximation ratios.     

The partition problem

In this paper we describe several forms of thek-partition problem and give integer programming formulations of each case. The dimension of the associated polytopes and some basic facets are identified. We also give several valid and facet defining inequalities for each of the polytopes.Partial Support from NSF Grants DMS 8606188 and ECS 8800281 is gratefully acknowledged.     

Equivalence classes of problem instances for a continuous-time lot sizing and scheduling problem

In the context of a single machine, multi-item, continuous-time lot sizing and scheduling problem with a production rate alternating between the two values zero and full production rate, this paper identifies classes of equivalent problem instances which differ from one another in the way in which demand is represented. Each class contains an instance with a continuous cumulated demand function, and may contain demand functions with very different shapes, including functions with discontinuities. When approaching the problem (in order to solve it numerically, or to prove analytical results etc.), it therefore becomes possible to select, from many different representations, the problem instance which best meets the requirements of the applied method.     

A branch-and-price approach for the partition coloring problem

This work proposes a new integer programming model for the partition coloring problem and a branch-and-price algorithm to solve it. Experiments are reported for random graphs and instances originating from routing and wavelength assignment problems arising in telecommunication network design. We show that our method largely outperforms previously existing approaches.     

The partition bargaining problem

Consider the problem of partitioning n items among d players where the utility of each player for bundles of items is additive; so, player r has utility for item i and the utility of that player for a bundle of items is the sum of the 's over the items i in his/her bundle. Each partition S of the items is then associated with a d-dimensional utility vector V^S whose coordinates are the utilities that the players assign to the bundles they get under S. Also, lotteries over partitions are associated with the corresponding expected utility vectors. We model the problem as a Nash bargaining game over the set of lotteries over partitions and provide methods for computing the corresponding Nash solution, to prescribed accuracy, with effort that is polynomial in n. In particular, we show that points in the pareto-optimal set of the corresponding bargaining set correspond to lotteries over partitions under which each item, with the possible exception of at most d(d-1)/2 items, is assigned in the same way.     

A graph partition problem

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The partition problem for finite abelian groups

By the aid of a slight generalization of the Hales-Jewett theorem [Trans. Amer. Math. Soc.106 (1963), 222–229] we investigate the partition problem for finite Abelian groups. In particular the partition problem for the class of finitely generated free modules over Z_q is solved. By the results of Deuber and Rothschild [“Coll. Math. Soc. János Bolyai 18,” 1976] this yields a complete characterization of those finite Abelian groups with respect to which the class FAB of all finite Abelian groups has the partition property. Especially it turns out that FAB has the partition property with respect to the cyclic group Z_m, m > 1.     

A matrix partition problem

In the set of positive definite semi-integral symmetric matrices we propose a partition problem. Then by introducing the notion of “additively prime” we obtain the generating function for this problem. Finally we establish the analyticity of the generating function.     

A subexponential algorithm for the coloured tree partition problem

Given a tree of n vertices and a list of feasible colours for each vertex, the coloured tree partition problem (CTPP) consists in partitioning the tree into p vertex-disjoint subtrees of minimum total cost, and assigning to each subtree a different colour, which must be feasible for all of its vertices. The problem is strongly NP-hard on general graphs, as well as on grid and bipartite graphs. This paper deals with the previously open case of tree graphs, showing that it is strongly NP-complete to determine whether a feasible solution exists. It presents reduction, decomposition and bounding procedures to simplify the problem and an exact algorithm of complexity (with ) for the special case in which a vertex of each subtree is given.     

On the modular sumset partition problem

A sequence m₁≥m₂≥?≥m_k of k positive integers isn-realizable if there is a partition X₁,X₂,…,X_k of the integer interval [1,n] such that the sum of the elements in X_i is m_i for each i=1,2,…,k. We consider the modular version of the problem and, by using the polynomial method by Alon (1999) [2], we prove that all sequences in Z/pZ of length k≤(p−1)/2 are realizable for any prime p≥3. The bound on k is best possible. An extension of this result is applied to give two results of p-realizable sequences in the integers. The first one is an extension, for n a prime, of the best known sufficient condition for n-realizability. The second one shows that, for n≥(4k)³, an n-feasible sequence of length k isn-realizable if and only if it does not contain forbidden subsequences of elements smaller than n, a natural obstruction forn-realizability.     

The satisfactory partition problem

The SATISFACTORY PARTITION problem consists in deciding if a given graph has a partition of its vertex set into two nonempty parts such that each vertex has at least as many neighbors in its part as in the other part. This problem was introduced by Gerber and Kobler [Partitioning a graph to satisfy all vertices, Technical report, Swiss Federal Institute of Technology, Lausanne, 1998; Algorithmic approach to the satisfactory graph partitioning problem, European J. Oper. Res. 125 (2000) 283-291] and further studied by other authors but its complexity remained open until now. We prove in this paper that SATISFACTORY PARTITION, as well as a variant where the parts are required to be of the same cardinality, are NP-complete. However, for graphs with maximum degree at most 4 the problem is polynomially solvable. We also study generalizations and variants of this problem where a partition into k nonempty parts (k?3) is requested.     

Solving huge size instances of the optimal diversity management problem

We report a real application project in a car industry optimization problem known as the optimal diversity management problem. We provide an alternative proof of NP-hardness, and we give and discuss the results obtained from a greedy algorithm applied to huge size instances.     

On the characterization and generation of nurse scheduling problem instances

Due to its complexity and relevance in practice, many different procedures have been proposed in the operations research literature to solve the well-known nurse scheduling problem (NSP). The NSP assigns nurses to shifts per day maximizing the overall quality of the roster while taking various constraints into account. The often highly case-specific workplace conditions in hospital departments have resulted in the development of dedicated (meta-)heuristics to find a workable schedule in an acceptable time limit. However, in spite of research community posing a growing need for benchmarking, these procedures lack any base for comparison.     

On solving large instances of the capacitated facility location problem

We present two sets of results pertaining to the solution of capacitated facility location problems that are large, especially with regard to the number of customers. One set of results relates to customer aggregation, while another set of results concerns the judicious selection of variable-upper-bounding (VUB) constraints to include in the initial integer-programming formulation.     

Worst-case analysis of the differencing method for the partition problem

We prove that the worst-case performance ratio of the Karmarkar-Karp differencing method for the Partition Problem is 7/6