Alexander Veremyev

Identifying large robust network clusters via new compact formulations of maximum k-club problems

Network robustness issues are crucial in a variety of application areas. In many situations, one of the key robustness requirements is the connectivity between each pair of nodes through a path that is short enough, which makes a network cluster more robust with respect to potential network component disruptions. A k-club, which by definition is a subgraph of a diameter of at most k, is a structure that addresses this requirement (assuming that k is small enough with respect to the size of the original network). We develop a new compact linear 0–1 programming formulation for finding maximum k-clubs that has substantially fewer entities compared to the previously known formulation (O(kn2) instead of O(nk+1), which is important in the general case of k>2) and is rather tight despite its compactness. Moreover, we introduce a new related concept referred to as an R-robust k-club (or, (k,R)-club), which naturally arises from the developed k-club formulations and extends the standard definition of a k-club by explicitly requiring that there must be at least R distinct paths of length at most k between all pairs of nodes. A compact formulation for the maximum R-robust k-club problem is also developed, and error and attack tolerance properties of the important special case of R-robust 2-clubs are investigated. Computational results are presented for multiple types of random graph instances.

Exact identification of critical nodes in sparse networks via new compact formulations

Critical node detection problems aim to optimally delete a subset of nodes in order to optimize or restrict a certain metric of network fragmentation. In this paper, we consider two network disruption metrics which have recently received substantial attention in the literature: the size of the remaining connected components and the total number of node pairs connected by a path. Exact solution methods known to date are based on linear 0–1 formulations with at least

Critical nodes for distance-based connectivity and related problems in graphs

\varTheta (n^3)

Detecting large cohesive subgroups with high clustering coefficients in social networks

Θ(n3) entities and allow one to solve these problems to optimality only in small sparse networks with up to 150 nodes. In this work, we develop more compact linear 0–1 formulations for the considered types of problems with

An integer programming approach for finding the most and the least central cliques

\varTheta (n^2)

Exact MIP-based approaches for finding maximum quasi-cliques and dense subgraphs

Θ(n2) entities. We also provide reformulations and valid inequalities that improve the performance of the developed models. Computational experiments show that the proposed formulations allow finding exact solutions to the considered problems for real-world sparse networks up to 10 times larger and with CPU time up to 1,000 times faster compared to previous studies.