G. Beretta | Searcharxiv

arXiv2024

On generalized KKT points for the Motzkin-Straus program

In 1965, T. S. Motzkin and E. G. Straus established an elegant connection between the clique number of a graph and the global maxima of a quadratic program defined on the standard simplex. Over the years, this seminal finding has inspired a number of studies aimed at characterizing the properties of the (local and global) solutions of the Motzkin-Straus program. The result has also been generalized in various ways and has served as the basis for establishing new bounds on the clique number and developing powerful clique-finding heuristics. Despite the extensive work done on the subject, apart from a few exceptions, the existing literature pays little or no attention to the Karush-Kuhn-Tucker (KKT) points of the program. In the conviction that these points might reveal interesting structural properties of the graph underlying the program, this paper tries to fill in the gap. In particular, we study the generalized KKT points of a parameterized version of the Motzkin-Straus program, which are defined via a relaxation of the usual first-order optimality conditions, and we present a number of results that shed light on the symmetries and regularities of certain substructures associated with the underlying graph. These combinatorial structures are further analyzed using barycentric coordinates, thereby providing a link to a related quadratic program that encodes local structural properties of the graph. This turns out to be particularly useful in the study of the generalized KKT points associated with a certain class of graphs that generalize the notion of a star graph. Finally, we discuss the associations between the generalized KKT points of the Motzkin-Straus program and the so-called replicator dynamics, thereby offering an alternative, dynamical-system perspective on the results presented in the paper.

math.OC↗

arXiv2023

Vector Flows and the Capacity of a Discrete Memoryless Channel

One of the fundamental problems of information theory, since its foundation by Shannon in 1948, has been the computation of the capacity of a discrete memoryless channel, a quantity expressing the maximum rate at which information can travel through the channel. In the literature, several algorithms were proposed to approximately compute the capacity of a discrete memoryless channel, being an analytical solution unavailable for the general discrete memoryless channel. This paper presents a novel approach to compute the capacity, which is based on a continuous-time dynamical system. Such a dynamical system can indeed be regarded as a continuous-time version of the Blahut-Arimoto algorithm. In fact, the updating map appearing in the Blahut-Arimoto algorithm is here obtained as a suitable discretization of the vector flow presented, using an analogy with some game-theoretical models. Finally, this analogy suggests a high-level hardware circuit design enabling analog computation to estimate the capacity.

cs.IT↗