The Maximum Edge Biclique Problem is NP-Complete

We prove that the maximum edge biclique problem in bipartite graphs is NP-complete.A biclique in a bipartite graph is a vertex induced subgraph which is complete.The problem of finding a biclique with a maximum number of vertices is known to be solvable in polynomial time but the complexity of finding a biclique with a maximum number of edges was still undecided.

Graph Parameters via Operator Systems

This work is an attempt to bridge the gap between the theory of operator systems and various aspects of graph theory. We start by showing that two graphs are isomorphic if and only if their corresponding operator systems are isomorphic with respect to their order structure. This means that the study of graphs is equivalent to the study of these special operator systems up to the natural notion of isomorphism in their category. We then define a new family of graph theory parameters using this identification. It turns out that these parameters share a lot in common with the Lov\'{a}sz theta function, in particular we can write down explicitly how to compute them via a semidefinte program. Moreover, we explore a particular parameter in this family and establish a sandwich theorem that holds for some graphs. Next, we move on to explore the concept of a graph homomorphism through the lens of C$^*$-algebras and operator systems. We start by studying the various notions of a quantum graph homomorphism and examine how they are related to each other. We then define and study a C$^*$-algebra that encodes all the information about these homomorphisms and establish a connection between computational complexity and the representation of these algebras. We use this C$^*$-algebra to define a new quantum chromatic number and establish some basic properties of this number. We then suggest a way of studying these quantum graph homomorphisms using certain completely positive maps and describe their structure. Finally, we use these completely positive maps to define the notion of a ``quantum" core of a graph.Mathematics, Department o

Semidefinite approximations for bicliques and biindependent pairs

We investigate some graph parameters asking to maximize the size of biindependent pairs (A,B) in a bipartite graph G=(V1∪V2,E), where A⊆V1, B⊆V2 and A∪B is independent. These parameters also allow to study bicliques in general graphs (via bipartite double graphs). When the size is the number |A∪B| of vertices one finds the stability number α(G), well-known to be polynomial-time computable. When the size is the product |A|⋅|B| one finds the parameter g(G), shown to be NP-hard by Peeters (2003), and when the size is the ratio |A|⋅|B|/|A∪|B| one finds the parameter h(G), introduced by Vallentin (2020) for bounding product-free sets in finite groups. We show that h(G) is an NP-hard parameter and, as a crucial ingredient, that it is NP-complete to decide whether a bipartite graph G has a balanced maximum independent set. These hardness results motivate introducing semidefinite programming bounds for g(G), h(G), and αbal(G) (the maximum cardinality of a balanced independent set). We show that these bounds can be seen as natural variations of the Lovász ϑ-number, a well-known semidefinite bound on α(G) (equal to it for G bipartite). In addition we formulate closed-form eigenvalue bounds, which coincide with the semidefinite bounds for vertex- and edge-transitive graphs, and we show relationships among them as well as with earlier spectral parameters by Hoffman, Haemers (2001) and Vallentin (2020)

Optimization-Based Network Analysis with Applications in Clustering and Data Mining

In this research we develop theoretical foundations and efficient solution methods for two classes of cluster-detection problems from optimization point of view. In particular, the s-club model and the biclique model are considered due to various application areas. An analytical review of the optimization problems is followed by theoretical results and algorithmic solution methods developed in this research. The maximum s-club problem has applications in graph-based data mining and robust network design where high reachability is often considered a critical property. Massive size of real-life instances makes it necessary to devise a scalable solution method for practical purposes. Moreover, lack of heredity property in s-clubs imposes challenges in the design of optimization algorithms. Motivated by these properties, a sufficient condition for checking maximality, by inclusion, of a given s-club is proposed. The sufficient condition can be employed in the design of optimization algorithms to reduce the computational effort. A variable neighborhood search algorithm is proposed for the maximum s-club problem to facilitate the solution of large instances with reasonable computational effort. In addition, a hybrid exact algorithm has been developed for the problem. Inspired by wide usability of bipartite graphs in modeling and data mining, we consider three classes of the maximum biclique problem. Specifically, the maximum edge biclique, the maximum vertex biclique and the maximum balanced biclique problems are considered. Asymptotic lower and upper bounds on the size of these structures in uniform random graphs are developed. These bounds are insightful in understanding the evolution and growth rate of bicliques in large-scale graphs. To overcome the computational difficulty of solving large instances, a scale-reduction technique for the maximum vertex and maximum edge biclique problems, in general graphs, is proposed. The procedure shrinks the underlying network, by confirming and removing edges that cannot be in the optimal solution, thus enabling the exact solution methods to solve large-scale sparse instances to optimality. Also, a combinatorial branch-and-bound algorithm is developed that best suits to solve dense instances where scale-reduction method might be less effective. Proposed algorithms are flexible and, with small modifications, can solve the weighted versions of the problems

Sum-of-squares representations for copositive matrices and independent sets in graphs

A polynomial optimization problem asks for minimizing a polynomial function (cost) given a set of constraints (rules) represented by polynomial inequalities and equations. Many hard problems in combinatorial optimization and applications in operations research can be naturally encoded as polynomial optimization problems. A common approach for addressing such computationally hard problems is by considering variations of the original problem that give an approximate solution, and that can be solved efficiently. One such approach for attacking hard combinatorial problems and, more generally, polynomial optimization problems, is given by the so-called sum-of-squares approximations. This thesis focuses on studying whether these approximations find the optimal solution of the original problem.We investigate this question in two main settings: 1) Copositive programs and 2) parameters dealing with independent sets in graphs. Among our main new results, we characterize the matrix sizes for which sum-of-squares approximations are able to capture all copositive matrices. In addition, we show finite convergence of the sums-of-squares approximations for maximum independent sets in graphs based on their continuous copositive reformulations. We also study sum-of-squares approximations for parameters asking for maximum balanced independent sets in bipartite graphs. In particular, we find connections with the Lovász theta number and we design eigenvalue bounds for several related parameters when the graphs satisfy some symmetry properties.<br/