The Paired Assignment Problem

We consider a variation of the maximum bipartite matching problem where each completed task must have at least two agents assigned to it. We give an integer programming formulation for the problem, and prove that the basic solutions of LP-relaxation are half-integral. It is shown that a fractional basic solution can be further processed to obtain an optimal solution to the problem.

• Matching Problems
• Linear Programming
• Basic Solutions
• Graph Algorithms

1. Introduction

1.1. Problem Definition

We are given a bipartite graph where each edge has one endpoint in and the other endpoint in. Elements of are normally referred as agents (or people), and elements of are referred as tasks (or jobs). Then means that agent can perform task (not every agent can perform every task). In classic maximum bipartite matching problem the goal is to find a matching in (a set of pairwise non-adjacent edges) that contains the largest possible number of edges. A matching is a one-to-one assignment: each agent can be assigned to at most one task, and each task can be assigned to at most one agent.

We consider the following variation of the maximum bipartite matching problem. Each agent still can be assigned to at most one task. But in our problem a task can be completed only if at least two agents are assigned to it. The goal is to maximize the number of completed tasks.

The problem can be given by the following integer program (IP):

Here is the set of agents, is the set of tasks. is a binary variable which is equal to 1 if task is completed. is a binary variable which is equal to 1 if agent is assigned to task. The objective of Function (1) is trying to maximize the number of completed tasks. Constraint (2) provides that each agent is assigned to no more than one task. Constraint (3) provides that if a task is completed then at least two agents are assigned to it.

1.2. Applications

The problem was first considered in [1] as a solution method for a combinatorial problem related to circuit reduction. [1] gave an integer program for the problem. In this paper we give a more efficient solution method based on the LP-relaxation of the integer program.

A few typical examples of the problem are given below. A group has members (agents) who should be assigned to projects (tasks). Each member can work only on some of the projects based on her/his qualifications. A project can be pursued only if at least two members are assigned to it. The goal is to maximize the number of projects that are pursued.

In a variation of the facility location problem, potential facility sites are the tasks, and demand points are the agents. Not every potential facility can serve every demand point (based on distance, compatibility, etc.). It is economical to open a facility only if it is assigned to serve at least two demand points. The goal is to maximize the number of open facilities.

Another possible situation is in the following. A company should assign guides to several tourist groups (tasks). Each group is from a certain country and needs guides who speak their language. The company has several guides (agents); each guide speaks several languages. Each group should be assigned two guides (primary and backup) satisfying the language requirement. The goal is to maximize the number of possible assignments.

1.3. Literature Review

Matching and assignment problems are of great importance in graph theory and combinatorial optimization ([2] [3] [4] ). The history of development, applications and solution methods of matching and assignment problems are discussed in [4] . Some variations of matching problems are discussed in [5] . A survey of assignment problems is given in [6] . In most variations the goal is to find a one-to-one assignment subject to some kind of restrictions. But some variations allow assignments of multiple agents to the same task or multiple tasks to the same agent ([7] -[9] ). The generalized assignment problem ([8] ) allows an agent to do multiple tasks provided that the set of tasks assigned to an agent do not exceed its capacity. In [7] an agent can be assigned several tasks, and the goal is to find an assignment that minimizes the total time of completing all the tasks.

Our model was introduced in [1] . To the best of our knowledge, no other model has considered the variation that a task can be completed only if two or more agents are assigned to it.

1.4. Our Results

The maximum bipartite matching problem can be solved by network flow techniques. It can be formulated as a maximum flow problem and solved by the augmenting path algorithm. Another solution method is linear programming. The constraint matrix of its integer program is totally unimodular, and thus the LP-relaxation returns integer solutions.

Those results do not extend to our problem. It is not clear how to use maximum flow techniques to solve the paired assignment problem. And as we show in Section 2, the constraint matrix of its integer program is not totally unimodular. But in the same section we show that any basic solution of the LP relaxation is half-integral; more specifically, each variable is integral, and each variable is half-integral. We use this special structure of basic solutions to design an algorithm that takes a half-integral basic solution as a starting point and gradually increases the number of completed tasks. The procedure to accomplish it is a modified version of breadth-first search. We prove that the algorithm returns an optimal solution for the paired assignment problem.

1.5. Outline of Paper

The paper is structured as follows. In Section 2, we show that any basic solution of the LP relaxation of (IP) is half-integral. In Section 3, we show how the basic solutions can be further processed to increase the number of completed tasks. In Section 4, we give an algorithm for solving the paired assignment problem and show that it returns an optimal solution. Future directions are discussed in Section 5.

2. Description of Basic Solutions of LP-Relaxation

The linear programming relaxation (LP) of the integer program (IP) is obtained by replacing the binary requirements of and with and .

Theorem 1 Basic solutions of (LP) are half-integral. Specifically, every variable is integer, and every variable takes value 0, 0.5, or 1.

Proof: Suppose the functional constraints of (LP) are rewritten in a standard form.

The coefficient matrix of Constraints (2) and (3) has the following form:

where

• represents the coefficients of variables in Constraints (2);

• represents the coefficients of variables in Constraints (3);

• represents the coefficients of variables in Constraints (3).

Matrix is totally unimodular because each column has exactly one 1 and one –1. Matrix is a diagonal matrix with 2’s on the main diagonal.

Suppose is a basis matrix for the augmented form of. Then the corresponding basic solution can be computed as follows:

.

Next we evaluate. First we expand by the columns of slack variables; in the result all the rows that correspond to basic slack variables will be crossed out. Then we expand by the columns of variables. Consider the following cases.

Case 1. Suppose we expand by a column of a variable that takes a fractional value in the basic solution. Then the slack variables of both and are basic, and thus both rows were crossed out in earlier expansions. So there is only one non-zero entry left in the column of, namely 2 in corresponding Constraint (3). Thus, the expansion will result in 2 times the corresponding cofactor.

Case 2. Suppose we expand by a column of a variable that takes value 1 in the basic solution. Then the slack variable of is basic, and thus the corresponding row was crossed out earlier. There are two non-zero entries in the column of: 2 in corresponding Constraint (3) and 1 in. The minor of entry 2 is 0 since after crossing its column only 0's are left in the row of. Thus, the expansion will include only one non-zero term, which is 1 times the corresponding cofactor.

Case 3. Suppose we expand by a column of a variable that takes value 0 in the basic solution. Then the slack variable of is basic, and thus the corresponding row was crossed out earlier. There are two non-zero entries in the column of: 2 in corresponding Constraint (3) and –1 in. The minor of entry 2 is 0 since after crossing its column only 0's are left in the row of. Thus, the expansion will include only one non-zero term, which is -1 times the corresponding cofactor.

The matrix obtained after crossing out all basic columns with corresponding rows is totally unimodular as a submatrix of a classic assignment problem. Thus, based on Cases (1)-(3), is either or where is the number of variables that take fractional values in the basic solution.

Next, for each different type of variable, we evaluate and the value of the variable.

Consider a basic variable.

Suppose is an element in the column of in a constraint of type (2) or (4). Then after crossing its column and row we still have 2’s in all basic -columns. Thus, the above analysis on still applies here, and the cofactor of is either or. Then the corresponding additive term in is 0, , or since we have only 0's and 1's in.

Suppose is an element in the column of in a constraint of type (3). The right-hand side of the constraint in the standard form is 0. Thus, the corresponding additive term in is also 0.

Summarizing, all the additive terms in are 0, , or. Thus, can take only integer values since is either or.

Consider a basic variable.

Suppose is an element in’s column. Submatrix obtained by crossing its column and row has only one 2 less than the original matrix. Thus, if we repeat the analysis done in Cases (1)-(3) for we can have the following possible values for its determinant: or if takes an integer value, or if takes a fractional value. Then the corresponding additive term in is 0, , , , or since we have only 0’s and 1’s in. Thus, the only values can take are 0, 0.5, and 1 since is either or. ,

3. Increasing Number of Completed Tasks by Reassignment

Suppose we have a solution to (LP). Based on Theorem 1, each variable takes one of the following values, 0, 0.5, or 1. Correspondingly, we distinguish three types of task-nodes in the current solution.

Definition 1 A task-node is called 

• completed if the corresponding;

• incomplete if the corresponding;

• unassigned if the corresponding.

Suppose we have a half-integral solution. It would be reasonable to include the completed tasks in the solution. But simply including the completed tasks might not give a good solution. Consider the example 1 of Figure 1. We will have the following convention for the rest of paper. Any arc that takes value 1 in the LP-relaxation will be called red arc and will be colored red (bold) in our figures; any arc that takes value 0 in the LP-relaxation will be called blue arc and will be colored blue in our figures. In the example of Figure 1, an optimal basic solution has no task with two agents assigned to it (both and are 0.5). But we can clearly have one completed task by assigning both and to as it is done in Figure 2.

In order to increase the number of completed tasks we need a reassignment from the current solution. The following result provides a general strategy for such a reassignment.

Lemma 1 Any reassignment that increases the number of completed tasks will decrease the number of incomplete tasks by at least two.

Proof: Suppose there are completed and incomplete tasks in the current solution with LP-value. Recall that is the optimal value of the LP-relaxation. Suppose there is another solution with at least completed tasks. Since the LP-value of any feasible solution cannot be more than then the number of incomplete tasks in the new solution is no more than. Lemma 1 implies that incomplete tasks should be the key in any reassignment that increases the number of

completed tasks. To get some insight how such an increase can be achieved consider example 2 given in Figure 3 and Figure 4 and example 3 given in Figure 5 and Figure 6.

In Figure 3, tasks and are incomplete, and task is completed. By reassigning agents, as it is done in Figure 4, becomes completed and unassigned, thus increasing the number of completed tasks from 1 to 2.

In Figure 5, tasks and are incomplete, task is completed, and is unassigned. By reassigning agents, as it is done in Figure 6, and become completed, and and become unassigned, thus increasing the number of completed tasks from 1 to 2.

We need the following definition to discuss the common pattern in the above examples.

Definition 2 Let and be two arcs with the same agent-node as origin. If in a current solution is red (assigned) and is blue (unassigned) then we call a red-blue arc pair.

All our examples, where we were able to create more completed tasks by reassignment, have the following feature. There are two incomplete tasks which are connected by a sequence of red-blue arc pairs. The number of completed tasks is increased by recoloring those red-blue pairs of arcs: the red arcs become blue, and the blue arcs become red. Recoloring the arcs essentially means reassigning every agent-node in the sequence to a different task.

In example 2, the original sequence is which becomes after recoloring. In example 3, the sequence is which becomes after recoloring.

The increase in number of completed tasks happens because the task-nodes in sequences change their statuses. In our examples, an unassigned task becomes assigned (e.g., task in example 3); incomplete nodes can become completed (e.g., task in example 2) or unassigned (e.g., task in example 2); completed tasks can become unassigned (e.g., task in example 3) or stay completed (e.g., task in example 2).

But we do not have a completed task-node with both incident arcs blue in the original sequence. It would mean that the two agents assigned to the task are not in the sequence. Thus, by recoloring we would assign two more agents to a task which is already completed. In that case it is unlikely that we would increase the number of completed tasks by reassignment.

The above analysis of the patterns observed in our examples leads to the following important concept.

Definition 3 A sequence of red-blue arc pairs, that connects two incomplete task-nodes, is called a valid path if

• there are no interior incomplete nodes on;

• any interior completed node on has at least one incident red arc in the sequence.

Using the concept of valid path, the following result generalizes the strategy of increasing the number of completed tasks observed in our examples.

Theorem 2 If there is a valid path connecting two incomplete task-nodes in a half-integral solution then we can increase the number of completed tasks by 1 by recoloring all the arcs on the path.

Proof: Let and be incomplete task-nodes that are connected by a valid path. We want to show that the number of completed tasks is increased by recoloring all the arcs on.

First we categorize the task-nodes on, and describe how recoloring will change their statuses.

1) If is an unassigned node then it has two incident blue arcs on. Thus, recoloring will make both arcs red, and will become a completed node (e.g., task in example 3).

2) If is an incomplete node and its incident arc on is blue then recoloring the arc will make completed (e.g., task in example 2).

3) If is an incomplete node and its incident arc on is red then recoloring the arc will make unassigned (e.g., task in example 2).

4) If is a completed node with two incident red arcs on then recoloring will make both arcs blue. Thus, will become an unassigned node (e.g., task in example 3).

5) If is a completed node with one incident red arc and one incident blue arc on then after recoloring will stay completed with one incident blue arc and one one incident red arc on (e.g., task in example 2).

Note that we cannot have a completed node with two incident blue arcs on since is a valid path.

As discussed above, only type (5) nodes do not change their status in the result of recoloring. Thus, our goal is to find out how status changes in other type of nodes affect the number of completed tasks on. To answer that question we need to discuss the possible configurations of type (1)-(4) nodes on.

Suppose and are task-nodes of type (1)-(4) on. We say and are -neighbors if all the internal task-nodes (if any) on the subpath of joining and are type (5) nodes. Based on the definition of type (5) nodes, any two neighboring arcs on the subpath joining two -neighbors have different colors. Also, based on their definitions, type (1) and (2) nodes have only incident blue arcs on while type (3) and (4) nodes have only incident red arcs on. Based on the last two observations, we have the following intermediate result.

Lemma 2 Any node of type (1) or (2) can be a -neighbor only with a node of type (3) or (4), and conversely, any node of type (3) or (4) can be a -neighbor only with a node of type (1) or (2). In other words, on, type (1) or (2) nodes are alternated by type (3) or (4) nodes.

Now we are ready to discuss how the number of completed tasks will be changed in the result of recoloring. We have three possible cases.

Case 1: Both incomplete nodes and on are of type 3. Based on Lemma 2, the number of type (1) nodes on is more than the number of type (4) nodes on exactly by 1. After recoloring, incomplete nodes and become unassigned, each type (1) (unassigned) node becomes completed, and each type (4) (completed) node becomes unassigned. Thus, the number of completed nodes is increased exactly by 1.

Case 2: Both incomplete nodes and on are of type 2. Based on Lemma 2, the number of type (4) nodes on is more than the number of type (1) nodes on exactly by 1. After recoloring, both and become completed, each type (1) (unassigned) node becomes completed, and each type (4) (completed) node becomes unassigned. Thus, the number of completed nodes is increased exactly by 1.

Case 3: One of and is of type 2, and the other one is of type 3. Based on Lemma 2, the number of type (4) nodes on is equal to the number of type (1) nodes on. After recoloring, the incomplete node which is of type (2) will become completed, each type (1) (unassigned) node becomes completed, and each type (4) (completed) node becomes unassigned. Thus, the number of completed nodes is increased exactly by 1.

Summarizing, in any case the number of completed nodes is increased exactly by 1. This concludes the proof of theorem 2. ,

4. Algorithm for Paired Assignment Problem

The result of Theorem 2 is the basis of the following algorithm for solving the paired assignment problem.

Algorithm 4.1 Algorithm for Paired Assignment Solve the LP-relaxation of the problem while there is a valid path connecting two incomplete task-nodes do recolor all the arcs along the valid path end while In the next two subsections we show that: 1) a valid path can be found efficiently using a modified version of breadth-first search (BFS); 2) the algorithm returns an optimal solution for the problem.

4.1. Procedure for Finding a Valid Path

We need to define an auxiliary digraph to do the search. For each red-blue pair of arcs we define a directed arc connecting the task-nodes by choosing the direction of the red arc. Let be the set of all directed arcs defined this way (note that two directed arcs and might have their original blue-red pairs sharing the red arc). Then we have a digraph defined on the set of all tasks. For example, the digraph corresponding to the original graph of Figure 5 is.

The search of a valid path can be done in. The equivalent of a valid path in is an undirected path connecting two incomplete nodes such that

• there are no interior incomplete nodes on;

• any interior completed node on has at least one of its incident arcs incoming.

The modified BFS for finding a valid path in is done as follows. One of the incomplete nodes, say, is chosen to be the root node. The modification to BFS concerns the completed nodes in the queue.

• If a completed node is reached from its parent-node in the queue through an incoming arc then is marked as fully visited and the search continues from as in standard BFS. Namely, after all the nodes reached from by an arc, incoming or outgoing, are included in the queue we dequeue and do not consider it again in the search (as suggested by its name).

• If a completed node is reached from its parent-node in the queue through an outgoing arc then is marked as partially visited. At this point, a node can be considered a child of and included in the queue only if it is reached from through an outgoing arc. But is not dequeued yet; we allow to visit it again. If at some point in the search it is visited from a node through an incoming arc then becomes fully-visited and the search from it is continued as in standard BFS described above.

We quit the search when 1) either another incomplete node is found; in this case the output is a valid path connecting and;

2) or no other incomplete node is found and there are only partially visited nodes left in the queue; in this case is not connected to another incomplete node by a valid path.

4.2. Optimality of the Algorithm Output

We claim that algorithm 4.1 returns an optimal solution for the original problem (IP), based on the following result.

Theorem 3 If there is no valid path connecting any two incomplete task-nodes then the number of completed tasks cannot be increased. Thus, Algorithm 4.1 returns an optimal solution for the paired assignment problem.

Proof: The proof is by induction on the number of task-nodes.

Basis step. The theorem statement is clearly true for any graph with only one task-node.

Inductive step.

Inductive hypothesis. Assume that the theorem statement is true for any graph with less than task-nodes. That is, for any graph with less than task-nodes, if

• assignment represents an optimal solution of (LP)• there is no valid path connecting any two incomplete task-nodes inthen has maximum possible number of completed tasks.

We need to prove the same for any graph with task-nodes. Suppose an instance of the problem is given by a graph with task-nodes. Let be a solution for• corresponding to an optimal solution of (LP)• and with no valid paths connecting incomplete nodes.

Let be a different solution that is obtained by reassigning some agents. Let the number of completed tasks in be. We need to prove that the number of completed tasks is no more than in.

If none of incomplete and unassigned tasks in becomes completed in then the number of completed tasks clearly cannot be increased. So we need to consider the following two cases.

1) An incomplete node in becomes completed in.

2) An unassigned node in becomes completed in.

Case 1: Node i is incomplete in S 1 and becomes completed in S 2 .

Get a reduced graph from by deleting task-node and the two agent-nodes and assigned to in. Denote the reduction of in by, and the reduction of in by.

Let be the node from which is getting its second assignment in. Namely, suppose task-node was assigned to in and is reassigned to in (see Figure 7). Note that cannot be an incomplete node in; otherwise would be a valid path connecting incomplete nodes and in. Thus, is a completed node in, and becomes incomplete in.

We can state the following about.

1) The number of completed tasks in is since it was in and was completed in but incomplete in.

2) is an optimal solution to the linear program for the reduced graph based on the following reason. We cannot have another assignment with larger LP-value for; otherwise, by adding and to, we could get an assignment with larger LP-value for original.

3) We cannot have connected to an incomplete node in by a valid path; otherwise would be a valid path connecting incomplete nodes and in. Thus, there are no valid paths connecting two incomplete task-nodes in since we didn't have any in and is the only new incomplete task in compared to.

Based on (2) and (3), the conditions for inductive hypothesis hold for. Thus, we can claim that the number of completed tasks cannot be increased in the result of reassignment from to. Then, based on (1), the number of completed tasks in is at most, and the number of completed tasks in is at most when we add to the completed tasks of.

Case 2: Node u is unassigned in S 1 and becomes completed in S 2 .

Get a reduced graph from by deleting task-node and the two agent-nodes and assigned to in. Denote the reduction of in by, and the reduction of in by.

Let and be the nodes from which is getting its assignments in. Note that and cannot be both incomplete in; otherwise we would have a valid path between and through the unassigned node (Figure 8). Then we might have the following two subcases.

Subcase 2.1: c is completed and d is incomplete in S₁.

Then, after deleting agent-nodes and with their assignments to and, becomes incomplete and becomes unassigned in.

We can state the following about.

1) The number of completed tasks in is since it was in and was completed in but incomplete in.

2) is an optimal solution to the linear program for the reduced graph based on the following reason. We cannot have another assignment with larger LP-value for; otherwise, by adding and to, we could get an assignment with larger LP-value for original.

3) We cannot have connected to an incomplete node by a valid path in; otherwise would be a valid path connecting incomplete nodes and in (see Figure 8). We also cannot have two incomplete nodes and connected by a valid path that has as an interior unassigned node in; otherwise would be a valid path connecting incomplete nodes and in (Figure 8). Thus, there are no valid paths connecting two incomplete nodes in.

Based on (2) and (3), the conditions for inductive hypothesis hold for, and we can claim that the number of completed tasks cannot be increased in the result of reassignment from to. Then, based on (1), the number of completed tasks in is at most. Hence the number of completed tasks in is at most when we add to the completed tasks of.

Subcase 2.2: Both c and d are completed in S₁.

Then, after deleting agent-nodes and with their assignments to and, both and become incomplete in.

We can state the following about.

1') The number of completed tasks in is.

2') is an optimal solution to the linear program for the reduced graph based on the following reason. We cannot have another assignment with larger LP-value for; otherwise, by adding and to, we could get an assignment with larger LP-value for original.

3') We cannot have both and connected to two different incomplete nodes and in by valid paths and; otherwise would be a valid path connecting incomplete nodes and in. But we might have one of and connected to an incomplete node in by a valid path, or and connected to each other by a valid path in.

Based on (3'), further division into cases is needed.

Subcase 2.2.1: Suppose none of and is connected to an incomplete node in by a valid path. Then there are no valid paths connecting two incomplete nodes in, and based on (2'), the conditions for inductive hypothesis hold for. Hence the number of completed tasks cannot be increased in the result of reassignment from to. Then, based on (1'), the number of completed tasks in is at most. Thus, the number of completed tasks in is at most when we add to the completed tasks of.

Subcase 2.2.2: Suppose is connected to an incomplete node in by a valid path (that incomplete node could be itself). Then after recoloring all the arcs on the path the number of completed tasks will increase by 1. Let be the solution obtained from by recoloring the arcs on the valid path.

We can state the following about.

1'') The number of completed tasks in is since it has one more completed task than.

2'') is an optimal solution to the linear program for the reduced graph since it is obtained from by recoloring the arcs on a valid path which is not changing the LP-value.

3'') There are no more valid paths joining two incomplete nodes in.

Based on (2'') and (3''), the conditions for inductive hypothesis hold for, and we can claim that the number of completed tasks cannot be increased in the result of reassignment from to. Then, based on (1''), the number of completed tasks in is at most. Hence the number of completed tasks in is at most when we add to the completed tasks of.

Thus, we proved that the number of completed tasks cannot be increased in any of the above cases. This completes the proof of Theorem 3. ,

5. Future Directions

Below are some possible future directions.

It would make sense to consider the weighted version of the problem. Weights could be associated with the arcs (to make it a variation of the assignment problem) or/and with the nodes.

It is interesting whether there is a purely combinatorial algorithm for solving the problem (without using linear programming).

A generalization of the paired assignemnt problem could be the following: task can be done only if at least agents are assigned to it.

• The special case when for every task is the classic maximum matching problem.

• Our problem is the special case when for every task.

• The problem for general might be hard to solve.

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