Elsevier

Discrete Optimization

Volume 35, February 2020, 100551
Discrete Optimization

Multiple Bipartite Complete Matching Vertex Blocker Problem: Complexity, polyhedral analysis and Branch-and-Cut

https://doi.org/10.1016/j.disopt.2019.100551Get rights and content

Abstract

Given a bipartite graph G=(UV,E), |U||V|, the surplus of G is defined by the maximum number k such that a matching covering all vertices of U still exists upon removal of any k vertices from V. Given a partition U={U1,,Um} of U, the Multiple Bipartite Complete Matching Vertex Blocker Problem (MBCMVBP) consists in finding a partition V={V1,,Vm} of V such that the smallest surplus among those of the induced subgraphs G[UiVi] is maximized. The removed vertices are related to the blocker notion. We show the strong NP-hardness of the MBCMVBP by using a reduction from the stable set problem. We also propose two integer linear programs for solving this problem. After comparing these two models, we introduce some valid inequalities for the model that outperforms the other one, and we analyze its facial structure. We then derive a Branch-and-Cut algorithm based on these results and conclude by an analysis of the experimental results.

Introduction

Solving interdiction/blocker problems is a mean to determine the maximum possible perturbation such that, beyond that point, no valid solution exists for the studied problem. Practically, these perturbations can be triggered by failures, attacks or absenteeism. Furthermore, such weakness analysis can be used to define tactical and strategic investments to make the model robust.

The matching interdiction/blocker problem [1], [2] consists in finding the maximum number of edges or nodes that can be removed such that a matching with a given cardinality still exists.

Given a bipartite graph G=(UV,E), |U||V|, the complete matching problem on U consists in finding a matching covering all vertices of U [3], [4]. The bipartite complete matching vertex blocker problem (BCMVBP) [5] consists in finding the maximum number k such that a complete matching on U still exists after removal of any k vertices in V. Let U={U1,,Um} be a partition of U. Given a partition V={V1,,Vm} of V, we denote by ki the solution of the BCMVBP associated with the induced subgraph G[UiVi],i{1,,m}. The Multiple Bipartite Complete Matching Vertex Blocker Problem (MBCMVBP) consists in finding a partition V={V1,,Vm} of V such that the minimum of {k1,,km} is maximized.

To the best of our knowledge, this problem has never been studied before. It can be applied in different fields. For instance, in nurse rostering and staff management, BCMVBP can be employed to identify the critical number of staff with specific skills below which the system remains under-staff. When considering several time shifts, the idea is to have the biggest common flexibility on each shift.

The complexity of the problem is analyzed, and two Integer Linear Programs (ILP) are proposed. Some polyhedral properties of the problem are identified including facets and valid inequalities to strengthen their efficiency in a Branch-and-Cut algorithm. All the proposed algorithms have been tested to evaluate the practical effectiveness of the algorithmic ingredients developed. Note that the instances are generated to represent real-life assignment problems such as nurse rostering.

The notion of interdiction/blocker is relevant in several fields. For example, considering computer networks, it is primordial to guarantee the continuity of services in case of dysfunction of some routers or switches. These problems are usually modeled by shortest paths [6], spanning trees [7] or network flows [8] in the underlying graph.

In the field of human resources, being able to anticipate the potential absenteeism is essential in order to ensure the proper running of a company. This class of problems can be formulated by matching or scheduling models.

The blocker problem allows the understanding of the strength of the graph according to a given property. It has been studied on several well-known graph properties such as the independent set [9], [10] and related structures, i.e. clique and chromatic number [11], [12] or path problem [13].

One of the problems related to the matching interdiction/blocker problem has been studied in [14], where the authors consider the minimum d-blocker problem. The goal is to determine a minimum cardinality subset of edges such that their deletion from the graph decreases the matching number by at least d units. This problem corresponds to a particular case of the edge interdiction problem.

In [1] and [11], the authors studied two particular versions of the matching interdiction problem. The first one, called edge interdiction problem, considers a graph in which each edge has a weight and an interdiction cost. The goal is to find a subset of edges such that the value of a maximum-weight matching without these edges is minimized, with respect to a cost constraint. The second problem is defined in a similar way, where costs are applied to vertices instead of edges. These problems are shown as being NP-complete even in bipartite graphs, and some approximation algorithms are proposed to solve them. The same author, in [2], extended his proof to show the NP-hardness of these problems in the particular scope of bipartite graphs.

In [5], the authors showed the polynomiality of this problem when a complete matching is considered. This problem is a particular case of the MBCMBVP when |U|=1. It has been applied to a robust nurse assignment problem which consists in affecting nurses to jobs according to their skills in such a way that possible absences may not perturb the schedule.

Note that these models can be applied to study problems of assignment in any domain where the management of staff or production with limited resources is needed (air or rail transports, timetabling, …).

In the next section, we give some basic definitions. In Section 3, the MBCMVBP is defined, and its NP-completeness is proved. In Section 4, we propose and study two Integer Linear Programs. The first one is based on the dual of the model used for the BCMVBP. The second one is an alternative with a reduced number of variables but an exponential number of inequalities. We also analyze the associated polytope and present a new family of inequalities. Then, some solving strategies and experimental results are presented in Section 5. In the last section, we propose some directions for further research.

Section snippets

Basic notions and definitions

Let us first introduce some notations and recall some basic notions concerning graphs, stable sets, and matchings.

Multiple Bipartite Complete Matching Vertex Blocker Problem (MBCMVBP)

First, we give some definitions, and we illustrate them on some examples. Then, we analyze the complexity of this problem.

Definition 1

Let G=(UV,E) be a bipartite graph, U={U1,,Um} be a partition of U and V={V1,,Vm} be a partition of V. For all iM={1,,m}, we denote by Hi the induced subgraph G[UiVi]. We say that (G,U,V) is k-Multiple Complete Matchable (k-MCM) on U if and only if, for all iM, Hi is k-CM on Ui.

The graph presented on Fig. 2 is 1-MCM on U: H1=G[U1V1] is 2-CM on U1 and H2=G[U2V2] is

Integer linear programs for MBCMVBP

In this section, we propose two models representing the MBCMVBP. The first model, called dual based formulation, is based on the integer linear program to solve the MBCMVBP when |U|=1. This problem has been introduced in [5] and will be called BCMVBP for short. The second model, called natural formulation, contains an exponential number of inequalities allowing to satisfy the property of Theorem 2, for all UiU,iM.

The comparison of these two models is presented in Section 5, where some

Implementation and experimental results

We first present the strategies used to solve our problems, explain how instances have been produced, and then analyze the obtained results.

Conclusion

We have studied the Multiple Bipartite Complete Matching Vertex Blocker Problem (MBCMVBP). Within the field of staff management, it can be used to identify the critical number of people below which the system becomes understaffed. The NP-completeness of the problem has been shown using a reduction from the stable set problem. Two mathematical formulations have been proposed. The first one is based on the dual formulation of a particular case and the second one (natural formulation) is based on

References (19)

There are more references available in the full text version of this article.

Cited by (0)

View full text