Discrete OptimizationA branch-and-cut algorithm for the Edge Interdiction Clique Problem
Introduction
Let be a simple undirected graph with vertices and edges. Two vertices and of are called neighbours if there is an edge and a subset of vertices is called a clique if they are all pairwise neighbours. The Maximum Clique Problem (MCP) asks for determining the largest clique of the graph whose size is denoted by the clique number of the graph. The MCP is one of the most studied problems in combinatorial optimization and graph theory. Many articles addressed it and we refer the interested reader to Coniglio, Furini, and San Segundo (2021), Li, Fang, Jiang, and Xu (2018a); Li, Liu, Jiang, Many, and Li (2018b), San Segundo and Tapia (2014), San Segundo, Coniglio, Furini, and Ljubi (2019a); San Segundo, Furini, and Artieda (2019b); San Segundo, Lopez, and Pardalos (2016b); San Segundo, Matia, Rodriguez-Losada, and Hernando (2013); San Segundo, Nikolaev, and Batsyn (2015); San Segundo, Rodríguez-Losada, and Jiménez (2011); San Segundo, Nikolaev, Batsyn, and Pardalos (2016c) where efficient exact algorithms are described also for some variants and generalizations of the problem. The MCP is strongly -hard and inapproximable in polynomial time to within any polynomial factor unless (Håstad, 1999).
In this paper, we address the Edge Interdiction Clique Problem (EICP) with the goal of developing an efficient exact algorithm to solve it to proven optimality. The EICP belongs to the family of problems aiming at reducing the clique number of the graph. Formally, given a graph and an interdiction budget the EICP asks to find a subset of at most edges to remove (also interdict) from so that the size of the maximum clique in the remaining graph is minimized. This problem, which has been introduced in Tang, Richard, and Smith (2016), is of practical importance for many applications arising in communication, social or biological networks in which cohesive clusters in the underlying network are represented as cliques.
As pointed out in Furini, Ljubić, Martin, and San Segundo (2019), clique interdiction problems play an important role in the context of crime detection and prevention. The authors of Berry et al. (2004) and Sageman (2004) identify large cliques as potential origins of catastrophic events such as terrorist or hacker attacks. As explained in Sageman (2004): “Dense networks like cliques commonly produce social cohesion and a collective identity and foster solidarity, trust, community, political inclusion, identity-formation, and other valuable social outcomes. Dense social networks foster intense face-to-face interactions in which collective identities are formed.” Consequently, in the analysis of terrorist networks (see, e.g., Chen, Chung, Xu, Wang, Qin, Chau, 2004, Sampson, Groves, 1989), cliques are used to model communities due to their ability to encode the interactions of groups of individuals who are all close friends with one another. Hence, the cohesiveness of a terrorist/criminal network, measured by the size of the maximum clique, is one of its important features which needs to be carefully monitored. The EICP application in this context consists of removing connections between the nodes of the terrorist network in order to reduce the most its cohesiveness.
Another important application of the EICP is in the context of image recognition. We refer the interested reader to e.g., San Segundo and Artieda (2015) and Stentiford (2014) and the references therein, for real-world applications of image recognition using maximum cliques. Specifically, maximum cliques are used to determine the maximum matching between two images. A set of key points for each image is determined via algorithms such as the Scale-Invariant Feature Transform (SIFT) algorithm or the Speeded-Up Robust Feature (SURF) algorithm (see Lowe, 2004 and Bay, Tuytelaars, & Gool, 2006). Each pair of key points, one from each image, becomes a vertex of the association graph. The edges of the association graph are pairs of vertices representing pairs of key points that can be matched in the two images. In this context, the maximum clique in the association graph represents the best matching between the two images. The solution of the EICP, with a given budget determines the set of edges which are crucial in the matching of the images. Moreover, by solving the EICP with different budget levels the optimal solution values provide a qualitative measure of the matching. If, by removing a small percentage of edges, the clique number in the resulting interdicted association graph largely decreases, then the matching is not of high quality. If, on the other hand, the value of the clique number in the resulting interdicted association graph remains high, then the matching can be considered of better quality.
In general, the EICP can be used to determine the resilience of a graph in terms of the potential of preserving its clique number after the interdiction of some of its edges. Formally, the edge clique-interdiction curve represents the decrease of the size of the maximum clique as a function of an incremental interdiction budget level. A similar curve has been introduced in Furini et al. (2019) where the interdiction is instead related to the vertices of the graph. In Section 7.7, we present the edge clique interdiction curves and analyze the resilience of several (social) networks.
The subset of edges that are interdicted from the graph constitutes an interdiction policy and defines the interdicted graph which corresponds to the original graph after the removal of the interdicted edges. Fig. 1 provides a graphical representation of the EICP and of its main features considering a synthetic example graph of 6 vertices and 13 edges. In this graph, the clique number is and there are four maximum cliques, i.e., (vertices depicted in grey), and . In Fig. 2(a) and (b), we report two optimal interdiction policies with and respectively. The edges belonging to the interdiction policies are depicted with red dashed lines. With the interdiction policy is composed of the edges and . After removing these edges, the clique number in the interdicted graph becomes . There are four remaining maximum cliques of size three, e.g., (vertices depicted in grey) or . With the interdiction policy is composed of the edges and . After removing these edges, the interdicted graph becomes triangle-free and its clique number is . There are eight remaining maximum cliques of size two, one of them is depicted with grey vertices, i.e., . It is important to notice that the optimal interdiction policy is often not unique. For instance, with another optimal interdiction policy is given by the edges and .
A trivial lower bound on the size of the clique in the interdicted graph is two, which is always obtained, unless all the edges of the graph are interdicted; without loss of generality, in the remainder of this article we assume . In the following section, we provide a comprehensive review of the literature addressing the EICP and its closely related problems.
Our research lies at the intersection of several related streams of literature: research on the edge interdiction clique problem, the node interdiction clique problem and the interdiction games. In this section, we review the key contributions of each of these streams and discuss how we extend them.
The EICP has been introduced by Tang et al. (2016). The authors introduced a generic exact approach for interdiction problems with binary leader variables and used the EICP as a special case study for testing their approach. Due to the generic nature of their method, the largest graphs considered in the computational study of Tang et al. (2016) contain at most 15 vertices, and the majority of them could not be solved within an hour of computing time. These instances have been solved to proven optimality by a recent a state-of-the-art exact solver for bilevel mixed integer programs provided in Fischetti, Ljubić, Monaci, and Sinnl (2019). In addition, a MIP-based heuristic for general interdiction problems has been proposed by Fischetti, Monaci, and Sinnl (2018), and larger EICP instances (ranging from 40 to 60 vertices) have been used to test the computational efficiency of their approach. Nevertheless, the question on how to design an efficient algorithm that can solve the EICP instances on large graphs relevant for real-world applications still remains an open issue. To the best of our knowledge, this work is a first study dedicated to the EICP in which a tight problem-specific ILP formulation is presented, along with efficient upper bounding techniques.
Another relevant stream of research is dedicated to the Node Interdiction Clique Problem (NICP) and its blocker-variant known as the Minimum Vertex Blocker Clique Problem (MVBCP). Furini et al. (2019) defined the NICP, in which for a given graph and an interdiction budget the decision maker has to find a subset of at most vertices to remove from so that the clique number in the remaining graph is minimized. The authors designed an exact algorithm CLIQUE-INTER, which is able to report optimal solutions for most instances on randomly generated graphs with up to 150 vertices. Optimal solution values are also reported for some larger networks from the SNAP database.1Pajouh, Boginski, and Pasiliao (2014) defined the MVBCP, in which one searches for a subset of vertices of minimum cardinality to be removed from a graph so that the maximum (weighted) clique in the remaining graph is bounded from above. The authors provided an analytical lower bound for the MVBCP and gave a MIP formulation with an exponential number of constraints, along with the characterization of conditions under which they are facet-defining. It is worth pointing out that the studies by Furini et al. (2019) and Pajouh et al. (2014) deal with the node interdiction aspect, and as such, cannot be straight-forwardly extended to the EICP. Similarities between the known results for the NICP and their exploitability in the EICP context are addressed in Section 3.
Finally, the NICP also belongs to a larger family of Interdiction Games under Monotonicity, which has been recently addressed by Fischetti et al. (2019). These problems involve two players, a leader and a follower, who play a Stackelberg game. The leader has a limited interdiction budget to remove a subset of items, whereas the follower solves a maximization problem based on the remaining items. The follower’s subproblem is assumed to satisfy a monotonicity property, which is then exploited for deriving a single-level integer linear programming reformulation. Whereas this monotonicity property is preserved by the NICP, it no longer holds for the EICP.
The main contribution of this manuscript is the development of a specialized exact algorithm to solve the EICP to proven optimality. To this end, we have developed an effective and new ILP formulation with an exponential number of constraints. We show that this formulation dominates the Benders-like problem reformulation, typically used for solving interdiction problems, and we develop a dedicated branch-and-cut algorithm. There are two key features of our new branch-and-cut algorithm: an efficient combinatorial algorithm which allows to perform the (-hard) separation of the inequalities in relatively short computing time; an effective initialization phase in which high-quality feasible solutions are computed thanks to a specialized heuristic algorithm. The specialized separation and the heuristic solutions are crucial to speed up the convergence of our new branch-and-cut algorithm as well as for reducing the exit gaps for the instances which are not solved to proven optimality.
The paper is structured as follows. In Section 2 a bilevel model is presented and in Section 3 we present a Benders-like problem reformulation. The first one can be directly solved by a general purpose bilevel solver, while the second one requires a branch-and-cut algorithm to be solved. In Section 4 we present the new set-covering-based ILP formulation which is the base of the new branch-and-cut algorithm developed in this paper. In the same section, we present and analyze the exponential-size family of inequalities called the clique-covering inequalities. Section 4 also contains a theoretical comparison of the strength of the clique-covering inequalities against the ones of the Benders-like reformulation. In Section 5 we present the combinatorial branch-and-bound algorithm used to effectively separate the clique-covering inequalities. This branch-and-bound algorithm is based on a state-of-the-art exact algorithm for the Maximum Clique Problem. In Section 6 we present and discuss the heuristic initialization algorithm which is composed of two main steps: the first step consists of finding a feasible solution solving a compact ILP formulation, and the second step is a specialized heuristic algorithm based on local-branching constraints. In Section 7 we present the computational results testing the performance of the newly developed branch-and-cut algorithm on several sets of benchmark instances. Finally, in Section 8 we present some concluding remarks and elaborate on future research directions.
Section snippets
A bilevel model of the EICP
Let be a vector of binary variables associated with the set of edges each variable encoding whether the corresponding edge is interdicted or not. Similarly, let be a vector of binary variables associated with vertices, indicating whether a vertex is part of a maximum clique in the interdicted graph. Then, the EICP problem can be stated as the following bilevel optimization problem:
The
Benders-like problem reformulation
Let represent the set of incidence vectors of all cliques in the graph i.e.:where the constraints ensure that two vertices cannot be part of a clique if there is no edge connecting them. Given an interdiction policy let be the associated set of interdicted (deleted) edges. We say that a clique of is interdicted (or, equivalently, covered) by if and only if .
We now provide a way to derive a Benders-like problem reformulation,
A set-covering problem reformulation
In this section we propose an alternative single-level problem formulation which uses set-covering arguments and requires an exponential number of constraints.
For let denote the set of cliques from whose size is equal to i.e., . Similarly, let denote the set of all cliques from whose size is such that .
Let and denote feasible lower and upper bounds to the optimal solution value of the EICP respectively, and let . To derive our
A combinatorial algorithm to solve the maximum clique separation problem
As explained in the previous section, to separate inequalities (5) and (7) it is necessary to compute a maximum clique in the interdicted graph derived from each interdiction policy . To solve the maximum clique problem to proven optimality we have customized the efficient combinatorial branch-and-bound (B&B) algorithm IMCQ employed for the node interdiction clique problem in Furini et al. (2019), which we will denote EIMCQ. One difference between IMCQ and EIMCQ is that the latter
Determining high-quality heuristic solutions
In this section, we describe the heuristic algorithm we designed to quickly obtain high-quality heuristic EICP solutions. Our approach is composed of two phases. The first one consists of an ILP-based heuristic which is described in Section 6.1. In this phase, an initial feasible solution is determined by solving a compact ILP model. This initial solution is then improved by a Local Branching phase which is described in Section 6.2. This second heuristic phase is based on a truncated version of
Computational results
In this section we assess the computational performance of the newly developed branch-and-cut algorithm which is called EDGE-INTER in the remainder of the paper. The purpose of this computational study is threefold: to evaluate the performance of EDGE-INTER determining, at the same time, the impact of its main features; to compare our exact algorithm against the state-of-art method available for general bilevel and interdiction problems from Fischetti, Ljubić, Monaci, and Sinnl (2017b)
Conclusions
In this manuscript we have addressed the Edge Interdiction Clique Problem (EICP), a very challenging problem from a computational perspective. The problem was introduced in Tang et al. (2016), and belongs to the family of interdiction problems. The aim of the EICP is to reduce as much as possible the clique number of a graph by removing a given number of edges.
The main goal of this paper was to design an effective exact algorithm to solve the EICP to proven optimality. To this end, we have
Acknowledgments
This work has been partially funded by the Spanish Ministry of Science, Innovation and Universities through the project COGDRIVE (DPI2017-86915-C3-3-R).
References (40)
- et al.
The generalized reserve set covering problem with connectivity and buffer requirements
European Journal of Operational Research
(2021) - DIMACS2 (2017) 2nd DIMACS implementation challenge – NP Hard Problems: Maximum Clique, Graph Coloring, and...
- DIMACS10 (2017). 10th DIMACS implementation challenge – Graph partitioning and graph clustering. URL...
- et al.
The maximum clique interdiction problem
European Journal of Operational Research
(2019) Clique is hard to approximate within
Acta Mathematica
(1999)- et al.
An exact approach for the vertex coloring problem
Discrete Optimization
(2011) - et al.
Minimum vertex blocker clique problem
Networks
(2014) - et al.
Improved initial vertex ordering for exact maximum clique search
Applied Intelligence
(2016) - et al.
A new exact maximum clique algorithm for large and massive sparse graphs
Computers & Operations Research
(2016) - et al.
Infra-chromatic bound for exact maximum clique search
Computers & Operations Research
(2015)