Discrete Optimization
Some constrained partitioning problems and majorization

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Abstract

We consider some constrained partitioning problems for a finite set of objects of different types. We look for partitions that are size- and type-similar, and, in addition, for a pair of such partitions that are “very different” in a certain sense. The motivation stems from a problem involving the partitioning of a set of students into smaller groups. We give these problems precise mathematical formulations and investigate these problems using the notion of majorization. A special case of one of the problems leads to a result concerning the packing of matchings in a bipartite graph.

Introduction

Let V be a finite set of N objects. Associated with each object is a certain type which is a number in {1,2,…,m}. We consider different problems involving partitions of V into subsets with certain properties. Each subset in a partition of V will be called a group. We require the groups to be size- and type-similar. Basically this means that each group contains about the same number of objects of each given type.

Before we define these problems in precise terms, we present the main motivation for the present study. The motivation springs from a problem one faces at the Faculty of Medicine at the University of Oslo, but we believe the problem should have a broader interest. In 1996 a new six year medical curriculum was implemented focusing on the integration of pre-clinical and clinical subjects, a high degree of self-regulated learning, early patient contact and improved skills and communication training. Traditional lectures were to a large extent replaced by problem-based learning (PBL) where six to eight students work together in a group with the guidance of a teacher. Simultaneously with this restructuring, a common curriculum was introduced for students of medicine, odontology and nutrition during their first two years of study. To develop the students' abilities to work with people of different professions in frequently changing settings, it was decided that the PBL-groups should be re-assembled in a special manner twice every term. The ideal goal was that both sexes and all three professions should be proportionally equally present in all groups and that no students who had previously worked together in a PBL-group should meet again. To begin with, the assembling of the groups was done manually with no particular algorithm, but the task was found both very time-consuming and unsatisfying. We were therefore consulted to develop a computerized method to perform the allocations.

The paper is organized as follows. Section 2 discusses the initial partitioning problem and an algorithm solving the problem is given. Section 3 describes the reallocation problem in terms of the notion of move matrices. Section 4 discusses a special case of the reallocation problem leading to a result concerning the packing of matchings in a bipartite graph. The general problem with solution algorithms are given in Section 5. We shall use some terms and results from majorization theory (see Section 2). For a comprehensive treatment of the theory and applications of majorization, see Marshall and Olkin [4]. We remark that many different partitioning problems have been investigated in the literature, many of these problems are NP-hard. We refer to Dahlhaus et al. [2] for a treatment of multiterminal cut problems and further references.

Section snippets

The initial partitioning problem

We want to partition the set of N objects into n groups with certain properties. We use the following notation:

    N:

    the number of objects, i.e., elements in V

    n:

    the number of groups

    m:

    the number of types

    di:

    the total number of objects of type i (im), so ∑idi=N

    d:

    d=(d1,d2,…,dm)

    pi,qi:

    defined (uniquely) by di=pi·n+qi, 0⩽qi<n

    :

    “mean of d”: d̄=(d1/n,d2/n,…,dm/n)


For a m×n matrix A we let rARm denote the row sum vector of A, i.e., rAi=∑j=1nai,j. Similarly, cARn denotes the column sum vector, so cAj=∑i=1mai

The reallocation problem

In this section we consider the second partitioning problem mentioned in Section 1. The problem is to re-assemble the PBL-groups of students to obtain new groups that are very different from the old groups. More precisely, one wants to minimize the number of revisits, where a revisit is a pair of students (or objects) being in the same groups in both assignments.

Note that when we are looking for a new assignment, only the type and old group of an object matter to us. Thus, we need not consider

A special case and packing of matchings

In this section we consider the strongly constrained reallocation problem (P2s) for the special case when di<n. Interestingly, this special case may be turned into a problem of packing perfect matchings on subgraphs of a complete bipartite graph.

Since di<n we have that pi=0 and qi=di for each i. Consequently, the matrix A1 constructed by Algorithm 1 is a (0,1)-matrix. Moreover, A1 has a very special structure which we call the consecutive increasing interval property (CII-property). This means

The weakly constrained reallocation problem

We now return to the weakly constrained reallocation problem (P2w). Consider a cyclic move matrix B (i.e. a move matrix which is cyclic). The first row of B contains p1 entries that are k1, followed by p2 entries that are k2 etc., and eventually pt entries that are kt. Here we may assume that k1,k2,…,kt are distinct numbers in {0,1,…,m} and p1+p2+⋯+pt=n. The matrix B therefore corresponds to moving pr objects of type kr (rt) from each old group in such a way that each new group receives pr

Concluding remarks

The algorithms have been implemented and tested on data from real student groups with success, reducing both planning time and the number of revisits compared to the ones obtained manually. The manual plans though consider some constraints not imposed in our model. For instance, due to timetable constraints the set of available groups may be reduced for some of the students.

As mentioned in the introduction the re-assembling of students happens more than once. A natural extension of the problem

Acknowledgements

The authors wishes to thank Per Grøttum for bringing these problems to our attention, and also for interesting discussions about the topic.

References (4)

  • R.K. Ahuja et al.

    Network Flows: Theory, Algorithms, and Applications

    (1993)
  • E. Dahlhaus et al.

    The complexity of multiterminal cuts

    SIAM Journal of Computing

    (1994)
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