Analysis of Integer Programming Model of Academic Load Distribution

Abstract

The bicriteria problem of academic load distribution (ALD) and its integer linear programming (ILP) model are considered. Earlier it was shown that the search for a feasible solution to this problem is NP-hard and the cardinality of the complete set of alternatives is polynomial. For the ALD problem, the problem of finding a Pareto-optimal solution can be formulated as a weighted bin packing problem with color constraints and lower bounds on the load of the bins. In this problem, the number of bins is given and items have volume and color. For each bin, there is an upper bound on the number of different colors and this bound depends on the bin volume. For each item, coefficients of the efficiency of placing in any bin are set. In this paper, we study the ILP model for finding a Pareto-optimal solution. Parametric families of ALD instances are constructed and the L-coverings of these instances are studied. These instances have a small duality gap, in particular, it can be equal to one. We investigate the complexity of solving these families by the Land and Doig algorithm for some known branching rules. It is shown, that the iterations number grows exponentially with the number of bins.

Supported by the program of fundamental scientific researches of the SB RAS No. I.5.1., project No. 0314-2019-0019.

1 Introduction

Distribution of academic load among teachers at an university department is one of the actual organization problems of the learning process. Problem formulations can be diverse and largely depend on the requirements of a particular university. In [1,2,3], some optimization models for this problem are proposed. In [1], the average load of a teacher at the department is minimized. The presented mixed integer programming model is a special variant of the fixed charge transportation problem. The NP-hardness of this problem is also shown. For the equivalent combinatorial statement of the problem, the branch and bound algorithm has been proposed there as well. However, the use of this model can lead to a solution whereby one course can be distributed arbitrarily among several teachers, which is contrary to the practice of Russian universities. In [2, 3], each training course consists of parts that correspond to some type of academic load, such as lectures, seminars, exams, etc. Each part is indivisible and can be assigned to only one teacher.

In [3], the bicriteria ALD problem was considered in which it is required to reduce the diversity of courses assigned to each teacher and to maximize the total effectiveness of the distribution. Shown that finding a feasible solution to this problem is NP-hard and the cardinality of the full set of alternatives is polynomial. Note that the bicriteria model of the ALD problem can be interpreted as a supply management problem with the discrete sizes of the batches [4].

Finding a Pareto-optimal solution to the ALD problem can be described using the terminology of the bin packing problem with color constraints. In considered problem, all items have volume and color and any bin has an upper and a lower bounds on total volume of items loaded into it. For any bin, there is an upper bound on the number of different colors. This bound is directly proportional to the volume of a bin. In addition, for each item, there is a set of the effectiveness coefficients (or weights) of placing it in any bin. Any placement of items is characterized by a total weight which should be maximized. Bin packing problems with color constraints are studied, for example, in [5, 6]. For these problems, the number of bins should be minimized in contrast to the considered problem in which the number of bins is given.

In this paper, we investigate the ILP model of finding a Pareto-optimal solution to the ALD problem. We propose two parametric families of instances that have the duality gap equal to one. It is shown that these families are difficult for the Land and Doig algorithm with some branching rules. Namely, it is shown, that the iterations number grows exponentially with the number of bins. On the basis of these families, a generalized family of instances for the ALD problem was constructed. These instances have a small duality gap and are difficult for the Land and Doig algorithm with the considered branching rules.

For the Land and Doig algorithm, families of difficult instances are known for the one-dimensional knapsack problem, the set covering problem, the set packing problem, the supply management problem, etc [7,8,9,10]. Analysis of difficult instances properties may be useful to increase the efficiency of integer programming algorithms. In addition, such families can be used for the theoretical study of other algorithms, for example, evolutionary ones [11].

To analyze the behavior of some other ILP algorithms when solving the proposed families, we study relaxation polytopes of proposed instances. We use the regular partitions approach, suggested by A.A. Kolokolov for analysis and solving the integer programming (IP) problems [12, 13]. It is based on using some special partitions of space $\mathbb {R}^n$, in particular, L-partition. The fractional covering of a problem is defined as a special subset of the relaxation polytope of the problem. All points of the fractional covering must be excluded from the relaxation polytope in a process of solving the problem when some well-known algorithms are applied. The L-covering cardinality is a characteristic of fractional covering of the IP problems and plays an important role in analysis of efficiency of many algorithms based on continuous optimization techniques. In particular, it is an upper bound on the iterations number for a class of cutting plane algorithms introduced in [13] which includes the first Gomory algorithm [14, 15].

The paper is organized as follows: in Sect. 2, we give the formulation of the ALD problem. In Sect. 3, we construct parametric families of instances for this problem and their generalization and study solving these families by the Land and Doig algorithm. In Sect. 4, we give the necessary information about the L-partition method and analyze the L-structure of the instances of constructed families.

2 Problem Formulation

The ALD problem and its model from [3] are described below. Let $I = \{1,...,m\}$ be a set of teachers. The values $a_i$, $c_i$ set the upper and the lower bounds of the possible load for teacher i where $i\in I$. Let $J = \{1,...,n\}$ be a set of training courses. We denote by $t_j$ a number of individual units of j-th course. Let $b_j^k$ be volume (i.e., the hours number) of k-th unit of j-th course where $k \in K_j = \{1, ..., t_j\}$.

We denote by $l_{ij}^{k}$ efficiency coefficient of assignment of unit k of course j to teacher i for all i, j, k. Let $s_j$ be a maximum number of units of one training course that can be assigned to one teacher ($s_j\le t_j$). Parameters $s_j$ are introduced to avoid the formation of monotonous load for the teachers.

It is necessary to assign each unit of any training course to a certain teacher so that a total load of each teacher satisfies the given bounds. The first goal of optimization is to minimize the number of the assigned courses for a teacher with the highest upper bound on his/her allowable load. For other teachers, an individual upper bound on the courses number is set. This value is proportional to the upper bound on the allowable load of the teacher. The second optimization goal is to maximize the total efficiency of assignments of units to the teachers.

We introduce Boolean variables $x_{ij}$ and $z_{ij}^k$ where $i \in I, \ \ j \in J, \ \ k\in K_j$. Here $x_{ij}= 1$ if teacher i gives course j and $x_{ij}= 0$, otherwise; $z_{ij}^k= 1$ if teacher i gives k-th unit of course j and $z_{ij}^k= 0$, otherwise. Denote the vector of variables $z_{ij}^k$ by Z and the vector of variables $x_{ij}$ by X.

Let $a_{max}=\max _{i\in I} a_i$ and $i_{max}$ be the index where this maximum is reached. Let $p_i=\frac{a_i}{a_{max}}$ for $i\in I$. We introduce a non-negative integer variable y. It is the upper bound on the number of courses assigned to teacher $i_{max}$.

The ALD problem can be formulated as a bicriteria ILP problem as follows

$$\begin{aligned} \text{ minimize } y \ \ \ \ \ \end{aligned}$$

(1)

$$\begin{aligned} \text{ maximize } L(Z)= \sum _{i=1}^{m} \sum _{j=1}^{n} \sum _{k=1}^{t_j} l_{ij}^{k}z_{ij}^{k} \ \ \ \ \ \end{aligned}$$

(2)

subject to

$$\begin{aligned} c_i\le \sum _{j=1}^{n} \sum _{k=1}^{t_j} b_j^k z_{ij}^{k}\le a_i, \ \ i \in I, \ \ \end{aligned}$$

(3)

$$\begin{aligned} \sum _{i=1}^{m} z_{ij}^k = 1, \ \ j \in J, \ k \in K_j, \end{aligned}$$

(4)

$$\begin{aligned} x_{ij}\le \sum _{k=1}^{t_j}z_{ij}^k \le s_j x_{ij}, \ \ i \in I, \ j \in J, \ \end{aligned}$$

(5)

$$\begin{aligned} \sum _{j=1}^{n} x_{ij} \le p_i y + q, \ \ i \in I, \end{aligned}$$

(6)

$$\begin{aligned} y \ge 0,\ y \in \mathbb {Z}, \ x_{ij}, z_{ij}^k \in \{0,1\}, \ i \in I, \ j \in J, \ k \in K_j. \end{aligned}$$

(7)

Conditions (3) describe the allowable ranges of the total load of each teacher. Equalities (4) show that each unit of any course should be assigned to just one teacher. Inequalities (5) describe the relationship of the variables $x_{ij}$ and $z_{ij}^k$. The variable $x_{ij}=1$ if and only if there exists index k such that $z_{ij}^k=1$, i.e., course j is assigned to teacher i if and only if at least one unit of this course is assigned to him/her. Restrictions (6) set the greatest number of courses for each teacher, depending on the value of his/her maximum permissible load. Here $q\in [0, \ 1)$ is a constant that controls the rounding of the values on the right side. Optimization criterion (1) means the minimization of the number of courses assigned to the teacher with the highest upper bound on allowable load. Criterion (2) maximizes the value of function L(Z), i.e., the total efficiency of the load distribution.

It was shown that finding a feasible solution of (1)–(7) is NP-hard [3].

The values of function (1) belong to set $\{1,2, ..., n\}$, i.e., the cardinality of the full set of alternatives does not exceed n.

Let integer parameter $y_{max}$ takes values from 1 to n. The problem of finding a Pareto-optimal solution is (2)–(7) with the additional constraint

$$\begin{aligned} y\le y_{max}. \end{aligned}$$

(8)

Notice that when problems (2)–(8) are being solved in order of decreasing of $y_{max}$, then some values $y_{max}$ can be skipped [4].

3 Analysis of Parametric Families of ALD Problem

We construct some families of instances of the ALD problem and study the branch and bound algorithm (Land and Doig scheme) for solving these instances.

3.1 Family F1(n)

Let $\alpha$, $\beta$ be positive integers and $\alpha \ge 3$. Consider the ALD instances with n teachers and n courses with the following input data. All courses consist of one unit of volume $\alpha$, i.e., $t_j=1$ for all j. Since index k takes a single value, we will omit it in the notation of variables and parameters, i.e., $b_j=b_j^1$, $l_{ij}=l_{ij}^{1}$, $z_{ij}=z_{ij}^1$ for all i, j. For teachers, the following lower and upper bounds on the possible load are set: $c_i=\alpha -1$, $a_i=\alpha$ for $i<n$ and $c_n=\alpha$, $a_n=\alpha +1$. Also, the efficiency coefficients are given: $l_{ij}=\beta$, $l_{nj}=\alpha +\beta$ for $i<n$ and all j. By F1(n) we denote the family of these instances.

From $s_j\le t_j$, it follows that $s_j=1$ and conditions (5) transform into $x_{ij}=z_{ij}$ for all i, j. Next, we replace the variables vector (Z, X) with $Z=(z_{ij})_{n \times n}$. Let $q\in [\frac{1}{\alpha }, \ 1)$ for $y_{max}=1$ and $q\in [0, \ 1]$, otherwise.

The problem of finding a Pareto-optimal solution may be written as

$$\begin{aligned} \text{ maximize } L(\bar{Z})= \beta \sum _{i=1}^{n-1} \sum _{j=1}^{n} z_{ij} + (\alpha +\beta )\sum _{j=1}^{n} z_{nj} \end{aligned}$$

(9)

subject to

$$\begin{aligned} \alpha -1\le \sum _{j=1}^{n}\alpha z_{ij}\le \alpha , \ \ i < n, \ \ \end{aligned}$$

(10)

$$\begin{aligned} \alpha \le \sum _{j=1}^{n}\alpha z_{nj}\le \alpha +1, \ \ \ \ \end{aligned}$$

(11)

$$\begin{aligned} \sum _{i=1}^{n} z_{ij} = 1, \ \ j \in J, \ \end{aligned}$$

(12)

$$\begin{aligned} z_{ij} \in \{0,1\}, \ i \in I, \ j \in J. \end{aligned}$$

(13)

In fact, constraints (6) are converted to

$$ \sum _{j=1}^{n} z_{ij} \le \frac{\alpha }{\alpha +1}y_{max} + q, \ \ i < n, $$

$$ \sum _{j=1}^{n} z_{nj} \le y_{max} + q. \ $$

For $y_{max}=1$, these constraints are a consequence of the right-hand inequalities from (10), (11). Therefore these restrictions are not included in the ILP model for F1(n). When $y_{max}>1$, this property is kept for all $q\in [0, \ 1)$.

Note that problem (9)–(13) can be considered as an instance of the generalized assignment problem with additional lower bounds on the total volume of items placed into each knapsack.

Denote by $Z^*$ and $\widetilde{Z}$ the optimal solutions of problem (9)–(13) and its linear relaxation which is obtained if restrictions (13) are replaced by $z_{ij}\ge 0$ for all i, j. From (10)–(12) it follows that the matrix $Z^*$ contains exactly one element equal to one in each row and each column. Another elements of $Z^*$ are zeros. The number of such solutions is n! and $L(Z^*)=n\beta +\alpha$.

The maximum of the objective function of the linear relaxation will be achieved when teacher n is fully loaded. Really this teacher has the maximum efficiency coefficients and the highest upper bound on the allowable load. The other teachers have efficiency coefficients equal to each other and with a lower value. Thus condition $\sum _{j=1}^{n}\tilde{z}_{nj}=1+\frac{1}{\alpha }$ is satisfied for any $\widetilde{Z}$. From (12), it follows that the sum of all $\tilde{z}_{ij}$ is equal to n. So there is the left-hand inequality from (10) which is satisfied as equality for some index i. For the remaining indexes, the right-hand inequalities are satisfied as equalities. Consequently, $\widetilde{Z}$ satisfies the conditions of a balanced transportation problem. If the continuous optimal solution $\widetilde{Z}$ is obtained by the simplex method, then it is a vertex of the polytope of the linear relaxation and this solution has the following property: it is impossible to construct a cycle for non-zero elements of the plan or for subsets of these elements.

The matrix $\widetilde{Z}$ can have a different structure. For any of them, condition $\sum _{j=1}^{n}\tilde{z}_{nj}=1+\frac{1}{\alpha }$ is satisfied and there are $j\in J$ such that $\tilde{z}_{nj}=\frac{1}{\alpha }$ and $i\not = n$ such that $\tilde{z}_{ij}=\frac{\alpha -1}{\alpha }$ (see Fig. 1). Clearly, $L(\widetilde{Z})=n\beta +\alpha +1$. The duality gap for the instances from F1(n) is equal to 1.

Fig. 1.

Examples of $\widetilde{Z}$ with a minimum and maximum number of non-integer elements.

3.2 Analysis of Land and Doig Algorithm

We study the branch and bound algorithm (Land and Doig scheme) for solving instances from F1(n). Denote this algorithm by LD. It is based on the sequential partition of the feasible set into subsets and on a calculation for each subset bounds on the objective function. These bounds are used to exclude subsets that do not contain optimal solutions for the integer problem. The bound is equal to the optimal value of the objective linear relaxation function and calculated by the simplex method. Algorithm LD constructs a binary search tree. Variable $z_{kl}$ corresponding to a fractional element of the optimal solution to the linear relaxation is chosen as a branching variable. At the current iteration, the subset of feasible solutions is divided into two subsets by adding one of the constraints $z_{kl}=0$ (left branch) or $z_{kl}=1$ (right branch). Such variables are called fixed, and other variables are called free. Different branching rules are applied when solving ILP problems (see, for example, [16,17,18]). The number of iterations of algorithm LD determined as the number of solved linear programming (LP) problems.

Let A1 be algorithm LD with a branching rule for which $k\not = n$ holds for any branching variable $z_{kl}$ when solving instances from F1(n). Suppose the variables of the problem are ordered as follows: $z_{11},..., z_{1n}, z_{21},..., z_{2n},..., z_{n1},..., z_{nn}$ then, for example, the rules are possible: the first fractional element, the fractional element that is closest to one, the fractional element with the lowest value $l_{ij}$.

Theorem 1

Algorithm A1 requires at least $2^n-1$ iterations to solve any instance from F1(n).

Proof

Let us analyze a part of the search tree that occurs during the running of the algorithm A1. We will be interested in branches (i.e. paths from the tree root to the current node) that have the length not less than $n- 1$. Each such branch corresponds to not less than $n-1$ fixed variables.

Let $z_{kl}=1$. From (10), it follows that variables $\tilde{z}_{kj}$ equal to 0 for $j \not =l$ in any feasible solution of the corresponding LP subproblem. From (12), it results that $\tilde{z}_{il} =0$ for $i\not =k$. So, variables of row k and column l of matrix Z cannot be selected below for branching. The LP subproblem has an integer optimal solution $Z^*$ and $L(Z^*)=n\beta +\alpha$ when assigning $n-1$ variables equal to ones. The tree branch breaks off.

Now we prove that all other nodes of the search tree at the level $n - 1$ have the upper bounds on the objective function equal to $L(\widetilde{Z})=n\beta +\alpha +1$.

Let $n-1$ variables from some row of the matrix Z be fixed at 0. Then we obtain an optimal continuous solution $\widetilde{Z}$, in which the single non-zero element in this row is equal to $\frac{a-1}{a}$ or 1. Assigning $n - 1$ zeros in some column j leads to an optimal continuous solution $\widetilde{Z}$ in which $\tilde{z}_{nj}=1$.

Suppose that $n - 1$ fixed zeros are located in arbitrary rows, excluding the row n. Note that for the linear relaxation of problem (9)–(13), the value $n (n - 1) (n - 1)!$ is the number of optimal solutions having the structure shown in Fig. 1 left. Here $n (n - 1)$ is the number of location variants of fractional elements and $(n - 1)!$ is the number of location variants of ones. If some variable $z_{kl} \ (k<n)$ is equal to 0, then $(n-1)!$ matrices $\widetilde{Z}$ of specified type are excluded from the relaxation set, namely, those for which $\tilde{z}_{kl}=\frac{\alpha -1}{\alpha }$. If $n-1$ zeros are fixed, then the number of excluded solutions is $(n-1)(n-1)!$. Therefore, there are at least n optimal solutions with the specified structure for such LP subproblem and $L(\widetilde{Z})=n\beta +\alpha +1$.

Consider the general case when t ones and $n - t - 1$ zeros are assigned where $1\le t \le n-2$. If t variables are fixed at 1 then the linear relaxation of problem (9)–(13) is converted to a similar LP subproblem of dimension $(n-t)\times (n-t)$. As follows from the case discussed above, the assignment of $n - t - 1$ zeros does not break the branch.

Thus, when solving instances from F1(n), algorithm A1 builds the search tree of depth at least $n - 1$, and this corresponds to solve at least $2^n-1$ LP problems. The theorem is proved.

Remark 1. Consider algorithm LD with any branching rule by which a branching variable is selected only from the last row of the matrix Z when solving instances from F1(n). As an example, we give the following rule: the fractional element with the highest coefficient $l_{ij}$. In this case, we can show that algorithm LD requires $n^2+n-1$ iterations to solve problems from F1(n).

3.3 Families F2(n) and G(2n)

For the ALD problem, consider the family F2(n) of instances which differ from the instances from F1(n) only by the parameters. We have $c_i=\alpha$, $a_i=\alpha +1$ for $i<n$ and $c_n=\alpha -1$, $a_n=\alpha$, $l_{ij}=\alpha +\beta$ and $l_{nj}=\beta$ for $i<n$ and all j. Here $\alpha \ge 3$, $\beta >0$ are integers.

The ILP problem to find a Pareto-optimal solution takes the following form

$$\begin{aligned} \text{ maximize } L(Z)= (\alpha +\beta ) \sum _{i=1}^{n-1} \sum _{j=1}^{n} z_{ij}+ \beta \sum _{j=1}^{n} z_{nj} \end{aligned}$$

(14)

subject to

$$\begin{aligned} \alpha \le \sum _{j=1}^{n}\alpha z_{ij}\le \alpha +1, \ \ i<n, \ \ \end{aligned}$$

(15)

$$\begin{aligned} \alpha -1\le \sum _{j=1}^{n}\alpha z_{nj}\le \alpha , \ \ \end{aligned}$$

(16)

$$\begin{aligned} \sum _{i=1}^{n} z_{ij} = 1, \ \ j \in J, \ \end{aligned}$$

(17)

$$\begin{aligned} z_{ij} \in \{0,1\}, \ i \in I, \ j \in J. \end{aligned}$$

(18)

Instances from F1(n) and F2(n) have the same set of optimal solutions. Linear relaxations of instances from F2(n) also have several optimal solutions, but these solutions have another structure. For example, one of the structures with the minimum number of fractional elements is shown in Fig. 2.

Fig. 2.

Example of structure of LP solution $\widetilde{Z}$ for instances of F2(n).

It is clear that $L(Z^*)=(n-1)\alpha +n\beta$, $L(\widetilde{Z})=(n-1)\alpha +n\beta +1$. The duality gap is still equal to one.

Denote by A2 the algorithm LD with a branching rule for which condition $k \not = n$ holds for any branching variable $z_{kl}$ when solving instances from F2(n). As examples of such rule, we can mention the following: the fractional element with the highest value of $l_{ij}$, the fractional element that is closest to zero or the first fractional element.

Theorem 2

Algorithm A2 requires at least $2^{n-1}$ iterations to solve any instance from family F2(n).

Proof

Initially, we analyze the optimal solution $\widetilde{Z}$ of the linear relaxation of problem (14)–(18) with $n-1$ fixed variables. Consider several cases.

Let $n-1$ variables from column $j_0$ of matrix Z be fixed to zeros. From (17) it follows that $z_{nj_0} = 1$, and from (16) it follows that $z_{nj} = 0$ for $j \not =j_0$. The other variables have the efficiency coefficients equal to $\alpha +\beta$. From here we obtain that $L(\widetilde{Z})=(n-1)(\alpha +\beta )+\beta$ for any feasible solution to the LP problem. If the optimal solution of the LP problem is integer, then this branch breaks off.

Let $n- 1$ zeros are fixed in several columns of matrix Z. Then there is an optimal solution $\widetilde{Z}$ with the minimum number of fractional components, and $L(\widetilde{Z})=L(Z^*)+1$. This can show as in the proof of Theorem 1.

Let a variable is fixed to 1, for example, $z_{kl}=1$ for $k\not = n$. From (17) it follows that $z_{il}=0$, $i\not = k$, and these variables will not be selected further for branching. Let $n-1$ ones are fixed. Then the linear relaxation has an optimal solution, which, up to a permutation of the columns, has the structure shown in Fig. 2. Solving the problem continues on this branch.

Consider the general case when t ones and $n-t-1$ zeros are assigned where $1\le t \le n-2$. For simplicity, we put $z_{kk}=1$ for $k \le t$. In addition, we assume that the other elements are equal to zeros in the first t rows of $\widetilde{Z}$, i.e., the corresponding variables are not selected for branching at the next iterations. In this case, the assignment of t ones means the transition to a subproblem of dimension $(n-t)\times n$. Let $z_{ij}=0$ for $i\le t$ and all j in this problem then we have the subproblem of type (14)–(18) of dimension $(n-t)\times (n-t)$. It can be shown as in the proof of Theorem 1 that the assignment of no more than $n-t-1$ zeros does not break this branch.

Therefore, all branches of the search tree have a length at least $n-1$. This tree has at least $2^n-1$ nodes, i.e., the number of the solved LP subproblems is at least $2^n-1$.

Now we only need to consider the case when the branching variable $z_{kl}$ is selected from the row in which some variable is already fixed to 1. If $z_{kl}=0$, then the LP subproblem is solvable and its dimension is keep and we have $L(\widetilde{Z})=L(Z^*)+1$. If $z_{kl}=1$ then the LP subproblem has no solution because a constraint from (15) is false with $i = k$. For the search tree of depth $n-1$, this means the insertion of the fragments corresponding to the just mentioned branchings. So, the number of iterations of algorithm A2 can only to increase.

The theorem is proved.

Remark 2. Consider algorithm LD with a branching rule, in which a branching variable is selected only from the last row of the linear relaxation solution to an instance from F2(n), for example, it is a fractional element that is closest to one or a fractional element with a minimum value of $l_{ij}$. Then, to solve instances from F2(n), algorithm LD requires at least $2n+1$ iterations.

Using F1(n) and F2(n), we construct a family G(2n). These instances have 2n teachers and 2n courses with a single unit of volume $\alpha$. Other input data is given below

$$ \begin{array} {ccccc} (l_{ij})_{2n\times 2n}=\left( \begin{array}{cc} L_1 &{} H \\ H &{} L_2 \end{array} \right)&\,&c=(c_1,c_2)&\,&a=(a_1,a_2). \end{array}$$

Here $L_1$, $L_2$ are the efficiency matrices of dimension $n\times n$; ${c}_1$, ${a}_1$ and ${c}_2$, ${a}_2$ are vectors of the lower and the upper bounds on the allowable load of teachers for families F1(n) and F2(n). H is a $(n\times n)$-matrix and all of its elements are equal to $-1$. For G(2n), the duality gap is equal to 2. Easy to show that this problem is difficult for algorithm LD with branching rules from algorithms A1 and A2.

4 Analysis of L-Covering

At first, we give the necessary information about the method of regular partitions [13]. Let $\mathbb {Z}^n$ be the set of all integer points of space $\mathbb {R}^n$. The L-partition of space $\mathbb {R}^n$ is defined as follows. Points $z\in \mathbb {Z}^n$ constitute the separate L-classes, i.e., the elements of partition. Points $x, \ y \not \in \mathbb {Z}^n$ $(x \succ y)$ belong to same fractional L-class if no $z\in \mathbb {Z}^n$ exist such that $x\succ z \succ y$ is holds. Here $\succ$ are the symbol of the lexicographical order. Denote by X/L the quotient set induced by L-partition for a set $X\subset \mathbb {R}^n$. Set X/L is called L-structure of set X and its elements are called L-classes. It is known that any fractional L-class V from X/L can be represented as:

$$\begin{aligned} V=X \cap \{x \ | \ x_{1}=d_{1}, ...,x_{r-1}=d_{r-1}, d_{r}<x_{r}<d_{r}+1\}, \end{aligned}$$

(19)

where $d_{j}$ is integer for $j=1, ...,r$ and $r \le n$.

We consider the following problem of finding the lexicographically maximal integer element of a set $\varOmega$

$$\begin{aligned} \text{ find }\quad \,\,\,\text{ lexmax } (\varOmega \cap \mathbb {Z}^n) \end{aligned}$$

(20)

where $\varOmega$ be some closed subset from $\mathbb {R}^n$.

Assume that the relaxation of this problem

$$ \text{ find } \quad \,\,\,\text{ lexmax } \ \varOmega $$

is solvable.

The fractional covering of problem (20) is defined as a set

$$\varOmega _*=\{ x \in \varOmega \ | \ x \succ z \text{ for } \text{ all } z \in ( \varOmega \cap \mathbb {Z}^n)\}.$$

The quotient set ${\varOmega _{*}/L}$ is called the L-covering of problem (20), and $|{\varOmega _{*}/L}|$ is called the L-covering cardinality.

Let us investigate the L-coverings of instances from family F1(n). Denote by M the polytope of the linear relaxation of an instance from F1(n). This polytope is described by conditions (10)–(12) and inequalities $z_{ij}\ge 0$ for all i, j. Vector Z denotes all variables $z_{ij}$ ordered in an arbitrary fixed way. As before, $Z^*$ and $\widetilde{Z}$ denote optimal solutions to the problem and its linear relaxation. Note that $L(\widetilde{Z})=L(Z^*)+1$. Let us introduce a new variable $z_0 = L(Z)$. It is clear that $z_0$ has integer value when vector Z is integer. Now we have the following lexicographical formulation of F1(n)

$$ \text{ find } \text{ lexmax }(\hat{M}\cap \mathbb {Z}^{n\times n+1}) $$

where $\hat{M}=\{(z_0,Z)\in \mathbb {R}^{n\times n+1} \ | \ z_0=L(Z), \ Z\in M\}.$

Let $\hat{M}_*$ be the fractional covering of this problem. In [14], the cardinality of the L-covering of the lexicographic optimization problem has been described through the L-structure of the relaxation polytope of the problem in the formulation with the objective function. We put $M_{L(\widetilde{Z})}=\{Z\in M \ | \ L(Z)=L(\widetilde{Z})\}$. For problem (9)–(13), the duality gap is equal to one and the lexicographical maximal element of set $\{Z\in M \ | \ L(Z)=L(Z^*)\}$ is integer. So above mentioned relation has the form

$$\begin{aligned} |\hat{M}_*/L|=|M_{L(\widetilde{Z})}/L|+1. \end{aligned}$$

(21)

The first sum term is the number of L-classes of set $\{Z\in \hat{M} \ | \ L(Z)=L(\widetilde{Z}\})$. The second term corresponds to the L-class with $L(Z^*)< z_0 < L(\widetilde{Z})$.

Theorem 3

When the order of variables is $z_{11}$, ..., $z_{1n}$, $z_{21}$, ..., $z_{2n}$, ..., $z_{n1}$, ..., $z_{nn}$, the following estimate holds for $n\ge 4$

$$|\hat{M}_*/L|\ge 1.7 \cdot n!.$$

Proof

As follows from (21), it suffices to estimate the number of optimal solutions of the linear relaxation of the problem (9)–(13) belonging to different L-classes. For simplicity, we will consider only solutions with the minimum number of fractional components. For a given order of variables, it is convenient to consider the solution Z as a matrix.

Property (19) of the L-partition implies that any fractional class is determined by the index r of its first fractional component and the values of the first $r-1$ integer components. Let the first fractional element of $\widetilde{Z}$ belongs to row i where $i <n$. According to Fig. 1, other elements of this row are equal to zeros, i.e., the number of variants of the fractional element location is equal to n. All variables of the preceding rows take integer values. Also from (10), it follows that each such row contains a single element equal to one and this element cannot belong to the same column as the first fractional element. Consequently, the number of such L-classes is equal to $P(i)= n (n-1) ... (n-i+1)$ for $i>1$ and $P(1)=n$. Thus, we have

$$ |\hat{M}_*/L|>|M_{L(\widetilde{Z})}/L|=\sum _{i=1}^{n-1}P(i)=n!\sum _{i=1}^{n-1}\frac{1}{(n-i)!}\ge 1.7\cdot n!. $$

The theorem is proved.

From this theorem, it follows that the upper bound on the number of iterations of a dual fractional cutting plane processes [13] including the first Gomory algorithm is exponential for family F1(n). The result holds for a lower bound on the number of iterations of such processes based on totally regular cuts [15]. Note that the ILP problems, possessing exponential L-coverings, are difficult also for the L-class enumeration method [19].

The following theorem is easy to prove.

Theorem 4

Consider the order of variables in which the variables $z_ {nj}$ are located in the first n places. Then the L-covering cardinality does not exceed $\frac{1}{2}(n-1)(n+2)$.

In this case the number of iterations for above mentioned cutting plane process is a polynomial.

Similar statements hold for the family F2(n).

5 Conclusion

We have considered the problem of academic load distribution among teachers as the bicriteria ILP problem and the single-criterion problem of searching for a Pareto optimal solution. We constructed parametric families of problems with duality gap equal to one and two. We showed that the proposed problems are difficult with certain branching rules for the Land and Doig algorithm, although with other branching rules the instances lose their complexity. The generalization of families remains difficult for the same branching rules. The study of the fractional covering of the problems showed that the cardinality of the L-covering varies depending on the order of variables from exponential to polynomial. This means that the constructed families of instances have the corresponding complexity for the L-class enumeration algorithm and some dual cutting plane algorithms. In the future, it is of interest to study more complex branching rules for the Land and Doig algorithm when solving the considered problem.