# title
A Lagrangian Bound on the Clique Number and an Exact Algorithm for the Maximum Edge Weight Clique Problem

# abstract
<jats:p> This paper explores the connections between the classical maximum clique problem and its edge-weighted generalization, the maximum edge weight clique (MEWC) problem. As a result, a new analytic upper bound on the clique number of a graph is obtained and an exact algorithm for solving the MEWC problem is developed. The bound on the clique number is derived using a Lagrangian relaxation of an integer (linear) programming formulation of the MEWC problem. Furthermore, coloring-based bounds on the clique number are used in a novel upper-bounding scheme for the MEWC problem. This scheme is employed within a combinatorial branch-and-bound framework, yielding an exact algorithm for the MEWC problem. Results of computational experiments demonstrate a superior performance of the proposed algorithm compared with existing approaches. </jats:p>

# authors[].organizations[].label
Texas A&M Engineering Experiment Station (TEES)

# publisher.label
Institute for Operations Research and the Management Sciences (INFORMS)

# keywords[]
Analytic Upper Bound On The Clique Number

# authorList[]
Hosseinian, S., Fontes, D., & Butenko, S.