Complexity And Heuristics For The Weighted Max Cut-Clique Problem

In market basket analysis (MBA), the goal is to understand human behavior to maximize sales. A clear behavior is buying correlated items. As a consequence, the determination of a set of items with a high correlation with others is a valuable tool for MBA. In this work, we address a combinatorial optimization problem with valuable applications to MBA, especially in marketing and product placement. The focus will be on the combinatorial problem and not on specific applications. For any given undirected graph G=(V,E) (where the nodes are items and the edges represent correlation), we aim to find the clique C subset of V such that the number of edges shared between C and V-C is maximized. This problem is known in the literature as the max cut-clique (MCC) problem. We can generalize this problem by considering the weights associated with each edge. In this context, we are interested in finding the clique C subset of V such that the weighted sum associated with each edge shared between C and V-C is maximized. The weighted version of the MCC problem is known as the maximum edge-weight neighborhood clique (MEWNC) problem. The main contributions of this paper are threefold. First, the computational complexity of both the MCC and MEWNC problems is established. Specifically, we prove that the MCC and MEWNC problems belong to the class of NP-complete problems. Second, an exact integer linear programming (ILP) formulation for the MEWNC problem is proposed. Third, a full greedy randomized adaptive search procedure/variable neighborhood descent methodology enriched with a tabu search is here developed, where the main components are novel neighborhood structures and a restricted candidate list that trade greediness for randomization in a multistart fashion. A dynamic tabu list considers a bounding technique based on the previous analysis. Finally, a fair comparison between our hybrid algorithm and a globally optimal solution using the ILP formulation confirms that a globally optimal solution is found by our heuristic for graphs with hundreds of nodes, but it is more efficient than the exact solution in terms of time and memory requirements.