Events

Title: Tails and Chains

Abstract:

Physicists are interested in "graphs with tails"; these are constructed by choosing a graph X and a subset C of its vertices, then attaching a path of length n to each vertex in C. We ask what is the spectrum of such graph? What happen if n increases? We will see that the answer reduces to questions about the matrix

\[ M(\zeta) := (\zeta_\zeta^{-1})I - A -\zeta D \]

where D is the diagonal 01-matrix with D_{i,i}=1 if i is in C.

Title: Approximation Algorithms for Matchings in Big Graphs

Abstract:

Matchings in graphs are classical problems in combinatorial optimization and computer science, significant due to their theoretical importance and relevance to applications. Polynomial time algorithms for several variant matching problems with linear objective functions have been known for fifty years, with important contributions from Tutte, Edmonds, Cunningham, and Pulleyblank (all with Waterloo associations), and they are discussed in the textbook literature.

Title: Spectral Properties of the exponential distance matrix

Abstract:

Given a graph $G$, the exponential distance matrix is defined entry-wise by letting the $(u,v)$-entry be $q^{dist(u,v)}$ where $dist(u,v)$ is the distance between the vertices $u$ and $v$ with the convention that if vertices are in different components, then $q^{dist(u,v)}=0$. We establish several properties of the characteristic polynomial (spectrum) for this matrix and the inertia of some graph families.

Title: Tightly-Secure Digital Signatures

Abstract:

This talk will cover two recent results on tightly-secure digital signature schemes from PKC 2021 and ASIACRYPT 2021.

The first part will consider a strong multi-user security definition for signatures with adaptive corruptions of secret keys, discuss its applications, and the technical challenges of constructing signatures with tight security in this setting. Then we will show a quite simple and efficient construction, based on sequential OR-proofs.

Title: Graph Continued Fractions

Abstract:

This talk is about a connection between matching polynomials and continued fractions. For the matching polynomials: we prove a refinement of a theorem by Ku and Wong, which extends the classical Gallai-Edmonds decomposition;

Title: Semidefinite Optimization Approaches for Reactive Optimal Power Flow Problems

Abstract:

The Reactive Optimal Power Flow (ROPF) problem consists in computing an optimal power generation dispatch for an alternating current transmission network that respects power flow equations and operational constraints. Some means of voltage control are modelled in ROPF such as the possible activation of shunts, and these controls are modelled using discrete variables. The ROPF problem belongs to the class of nonconvex MINLPs, which are NP-hard problems. We consider semidefinite optimization approaches for solving ROPF problems and their integration into a branch-and-bound algorithm.

Title: Structured (In)Feasibility: Nonmonotone Operator Splitting in Nonlinear Spaces

Abstract:

Several challenging optimization problems in power networks involve operational decisions, non-linear models of the underlying physics described by the network as well as uncertainty in the system parameters. However, these networks exhibit a nice structure. This talk provides an overview of approaches that combine recent advances in robust optimization and conic relaxations of polynomial optimization problems along with exploiting the structure of the underlying problem. These approaches are demonstrated on applications arising in power networks.

Title: Forcing Quasirandomness in Permutations

Abstract:

A striking result in graph theory is that the property of a graph being quasirandom (i.e. resembling a random graph) is characterized by the number of edges and the number of 4-cycles being close to the expected number in a random graph. Král’ and Pikhurko (2013) proved an analogous result for permutations; i.e. that quasirandom permutations are characterized by the densities of all permutations of length 4.

Title: Quantum information science for combinatorial optimization

Abstract:

Due to input-output bottlenecks, quantum computers are expected to be most applicable to problems for which the quantity of data specifying the instance is small but the computational cost of finding a solution is large. Aside from cryptanalysis and quantum simulation, combinatorial optimization provides some of the best candidates for problems of real-world impact fitting these criteria. Many of these problems are NP-hard and thus unlikely to be solvable on quantum computers with polynomial worst-case time complexity.

Title: Four families of polynomials in spectral graph theory

Abstract:

 In this talk we describe four families of polynomials related to the spectrum of a graph. First, some known main results involving such polynomials, such as the spectral excess theorem characterizing distance-regularity, are discussed. Second, some new results giving bounds for the $k$-independence number $\alpha_k$ of a graph are presented. In this context, we comment on some relationships between the inertia (Cvetkovi\`c) and ratio (Hoffman) bounds of $\alpha_k$.