Minimum cost edge blocker clique problem
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Minimum cost edge blocker clique problem Foad Mahdavi Pajouh1 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Given a graph with weights on its vertices and blocking costs on its edges, and a user-defined threshold τ > 0, the minimum cost edge blocker clique problem (EBCP) is introduced as the problem of blocking a minimum cost subset of edges so that each clique’s weight is bounded above by τ . Clusters composed of important actors with quick communications can be effectively modeled as large-weight cliques in real-world settings such as social, communication, and biological systems. Here, we prove that EBCP is NP-hard even when τ is a fixed parameter, and propose a combinatorial lower bound for its optimal objective. A class of inequalities that are valid for the set of feasible solution to EBCP is identified, and sufficient conditions for these inequalities to induce facets are presented. Using this class of inequalities, EBCP is formulated as a linear 0–1 program including potentially exponential number of constraints. We develop the first problem-specific branch-and-cut algorithm to solve EBCP, which utilizes the aforementioned constraints in a lazy manner. We also developed the first combinatorial branch-and-bound solution approach for this problem, which aims to handle large graph instances. Finally, computational results of solving EBCP on a collection of random graphs and power-law real-world networks by using our proposed exact algorithms are also provided. Keywords Edge blocker · Maximum weighted clique · NP-hard · Exact algorithms · Network interdiction
1 Introduction Network models of real-world systems often possess important properties that directly affect their functions, and as a result, should be preserved. One of the main drivers of such important properties is the network’s structure. In practice, one often needs to detect network components (e.g., vertices or edges) that play important roles in sustaining a desirable property. In related literature, some researchers assumed that a user-defined threshold for the property of interest is given, and the goal is to find a minimum size (cost) set of vertices or edges to block, and bound the property by the given threshold. This problem is known as the min-
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Foad Mahdavi Pajouh [email protected] Department of Management Science and Information Systems, University of Massachusetts Boston, Boston, MA 02125-3393, USA
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Annals of Operations Research
imum (cost) vertex/edge blocker problem (Bazgan et al. 2011; Ries et al. 2010; Zenklusen et al. 2009; Mahdavi Pajouh et al. 2014, 2015). Alternatively, one may need to identify a set of vertices or edges within a given size (budget) to block, and cause the largest change in the considered network property. This problem is recognized as the most vital vertices/edges problem (Bazgan et al. 2011; Bar-Noy et al. 1995; Bazgan et al. 2010). This paper focuses on the property of the maximum weight of a clique in a weighted network. We assume that we are only able to block
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