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

Given a simple graph and a constant $$\gamma \in (0,1]$$ γ ∈ ( 0 , 1 ] , a $$\gamma $$ γ -quasi-clique is defined as a subset of vertices that induces a subgraph with an edge density of at least $$\gamma $$ γ . This well-known clique relaxation model arises in a variety of application domains. The maximum $$\gamma $$ γ -quasi-clique problem is to find a $$\gamma $$ γ -quasi-clique of maximum cardinality in the graph and is known to be NP-hard. This paper proposes new mixed integer programming (MIP) formulations for solving the maximum $$\gamma $$ γ -quasi-clique problem. The corresponding linear programming (LP) relaxations are analyzed and shown to be tighter than the LP relaxations of the MIP models available in the literature on sparse graphs. The developed methodology is naturally generalized for solving the maximum $$f(\cdot )$$ f ( · ) -dense subgraph problem, which, for a given function $$f(\cdot )$$ f ( · ) , seeks for the largest k such that there is a subgraph induced by k vertices with at least f(k) edges. The performance of the proposed exact approaches is illustrated on real-life network instances with up to 10,000 vertices.