A Tutorial on Clique Problems in Communications and Signal Processing Ahmed Douik, Student Member, IEEE, Hayssam Dahrouj, Senior Member, IEEE, Tareq Y
1 A Tutorial on Clique Problems in Communications and Signal Processing Ahmed Douik, Student Member, IEEE, Hayssam Dahrouj, Senior Member, IEEE, Tareq Y. Al-Naffouri, Senior Member, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE Abstract—Since its first use by Euler on the problem of the be found in the following reference [4]. The textbooks by seven bridges of Konigsberg,¨ graph theory has shown excellent Bollobas´ [5] and Diestel [6] provide an in-depth investigation abilities in solving and unveiling the properties of multiple of modern graph theory tools including flows and connectivity, discrete optimization problems. The study of the structure of some integer programs reveals equivalence with graph theory the coloring problem, random graphs, and trees. problems making a large body of the literature readily available Current applications of graph theory span several fields. for solving and characterizing the complexity of these problems. Indeed, graph theory, as a part of discrete mathematics, is This tutorial presents a framework for utilizing a particular particularly helpful in solving discrete equations with a well- graph theory problem, known as the clique problem, for solving defined structure. Reference [7] provides multiple connections communications and signal processing problems. In particular, the paper aims to illustrate the structural properties of integer between graph theory problems and their combinatoric coun- programs that can be formulated as clique problems through terparts. Multiple tutorials on the applications of graph theory multiple examples in communications and signal processing. To techniques to various problems are available in the literature, that end, the first part of the tutorial provides various optimal e.g., [8]–[15].
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