Future students

Title: NC algorithm to find perfect matching in planar graphs

Abstract: Is perfect matching in NC? That is, is there a deterministic fast parallel algorithm for it? This has been an outstanding open question in theoretical computer science for over three decades, ever since the discovery of RNC matching algorithms. Within this question, the case of planar graphs has remained an enigma: On the one hand, counting the number of perfect matchings is far harder than finding one (the former is #P-complete and the latter is in P), and on the other, for planar graphs, counting has long been known to be in NC whereas finding one has resisted a solution. 

Title: The spectra of lift and factored lifts of graphs or digraphs

Abstract: In this talk, we first explain the path we took from defining polynomial matrices to a generalization of voltage graphs that we call combined voltage graphs. Moreover, we give a general definition of a matrix associated with a combined voltage graph, which allows us to provide a new method for computing the eigenvalues and eigenspaces of such graphs. 

Title: Nonnegative Matrix Factorization in non-standard settings, for fun

Abstract: This abstract is broken into 6 points.

1. What is NMF: NMF is to find two elementwise nonnegative low-rank matrices W and H such that M ≈ WH for a given elementwise nonnegative matrix M.

2. NMF is commonly done in the Euclidean distance.

3. I argue that Euclidean distance is "not good", and a ray-to-ray distance is better.

4. Under L2-normalization onto the unit sphere, we arrive at the cosine angle distance, which motivates the use of manifold optimization techniques.

5. Under L1-normalization onto the simplex, we arrive at the so-called Aitchison geometry, which contains funny algebra.

6. Why do these: for curiosity and fun. For application, these non-standard NMF can "remove cloud" from satellite images.

Title: Quartic quantum speedups for planted inference

Abstract: Consider the following task ("noisy 4XOR"), arising in CSPs, optimization, and cryptography.  There is a 'secret' Boolean vector x in {-1,+1}^n.  One gets m randomly chosen pairs (S, b), where S is a set of 4 coordinates from [n] and b is x^S := prod_{i in S} x_i with probability 1-eps, and -x^S with probability eps.  Can you tell the difference between the cases eps = 0.1 and eps = 0.5? 

Title: Oddities in the pursuit of self-duality

Abstract: Certain important cones in conic optimization are self-dual, e.g., the positive semidefinite matrices and the nonnegative orthant. In this talk we will address the problem of determining which convex cones are self-dual in a broad sense, which allows changing the underlying inner product if necessary. We will describe some recent progress on this question for polyhedral cones and discuss some unexpected connections. In particular, we will show a curious result connecting self-dual polyhedral cones and extreme rays of doubly nonnegative matrices. We will also discuss how semidefinite programming can be used to search for self-dual polyhedral cones. Time allowing, we will describe new results on the related problem of determining the automorphisms of certain cones of interest.

This talk is based on joint works with João Gouveia and Masaru Ito.

Title: Bivariate P-polynomial Association Schemes on the Fibers of Uniform Posets

Abstract: Orthogonal polynomials emerging in the context of P- and Q-polynomial association schemes are known to reside within the Askey-scheme. This relationship forms a bridge between algebraic combinatorics and the study of special functions, yielding significant benefits for both fields. Recent research has introduced multivariate generalizations of P- and Q-polynomial association schemes and provided numerous examples. This development aims notably to deepen our understanding of multivariate analogues of Askey-Wilson polynomials. In this talk, we will review this generalization, its connection to m-distance-regular graphs, and an algebraic combinatorial structure known as a "uniform poset," from which many examples of bivariate schemes appear to originate.

Title: Intersections of graphs and χ-boundedness: characterizing χ-guarding graph classes

Abstract: For two graphs G1, G2, their intersection is given by only keeping the vertices and edges that appear in both. This graph operation is closely related to various intersection graph classes, such as the intersection graphs of axis-aligned rectangles. We are interested in the interplay between the graph intersection operation and χ-boundedness. A graph class C  is χ-bounded if there exists a function providing an upper bound for the chromatic number of each graph in the class based on the graph’s clique number.

Title: Tight bounds for reconstructing graphs from distance queries

Abstract: Suppose you are given the vertex set of a graph and you want to discover the edge set. An oracle can tell you, given two vertices, what the distance is between these vertices in the graph. (For example, in a computer network, this would represent the minimum number of communication links needed to send a message from one computer to another.) Based on the answer, you may select the next query. The (labelled) graph is reconstructed when there is a single edge set compatible with the answers. How many queries are needed, in the worst case? The question becomes interesting for bounded degree graphs. We provide tight bounds for various classes of graphs, improving both the lower and upper bound, in both the randomized and deterministic setting. 

Title: Two variations of the chromatic symmetric function

Abstract: The /chromatic symmetric function/ is a symmetric function generalization of the chromatic polynomial that encodes the ways to color a graph such that no two adjacent vertices get the same color. We will discuss two different analogues of the chromatic symmetric function: a K-theoretic analogue called the /Kromatic symmetric function/, and a categorification called the /chromatic symmetric homology/. We show that certain properties of a graph can be recovered given its Kromatic symmetric function, and we give some formulas for special cases of the chromatic symmetric homology.

Title: Approximating Graphic TSP with matchings

Abstract: Revisiting defining character of the field of algorithm design and complexity, TSP!. While the problem itself is NP-Hard and difficult to approximate, various formulations, such as the Metric version, have yielded notable approximation algorithms. The classical 1.5-approximation algorithm by Christofides, leveraging matchings, stood as the best-known result for decades. However, recent breakthroughs have pushed these boundaries further.