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 01 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.
robust network clusters k-clubs r-robust k-clubs combinatorial optimization graph theory compact 01 formulations