Events

Title: Odd cycle transversal in P5-free graphs

Abstract: Odd cycle transversal is a fun computational problem, somewhere between colouring and independent set: we are (equivalently) looking for a bipartite induced subgraph of maximum weight. As one might expect, this is NP-hard; I will tell you how to solve this problem in polynomial time in P5-free graphs (and more). Joint work with Cece Henderson, Evelyne Smith-Roberge, and Rebecca Whitman.

Note: I am COVID-cautious and will bring masks for those willing to wear them. 

Title: More Martin and c2 details.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm.  Please also come to the pre-seminar to discuss what sorts of snacks and refreshments we might want going forward.

Abstract:  I'm going to tell you more of the details behind my Martin polynomial work last year with Erik Panzer which led to the proof of the c2 completion conjecture. I will actually describe the key bijection that proves it all, say some things about the permanent invariant, and cover other details that there hadn't been time for in the colloquium level presentation of the result.

Title: A O(log log (rank) ) - competitive algorithm for the matroid secretary problem

Abstract: In the Matroid Secretary problem the weighted elements of a matroid arrive one by one in a uniformly random order where an online algorithm observes the value of the element and must make an irrevocable decision of whether or not to include the element in its solution before the arrival of the next element. The goal is to maximize the total value of the chosen elements under the condition that they must constitute an independent set. Other than knowing the cardinality of the ground set and having access to an independence oracle, the algorithm has no further information about the matroid.

Title: The limits of the inertia bound

Abstract: Spectral graph theory provides us with a wide array of surprising results which relate graph-theoretic parameters to linear-algebraic parameters of associated matrices. Among the most well-known and useful of these is Hoffman’s ratio bound, which gives an upper bound on the independence number of a graph in terms of its eigenvalues.

Title: Intersections of graphs and χ-boundedness: Interval graphs, chordal graphs, and χ-guarding graph classes

Abstract: Following A. Gyárfás (1987), we say that a hereditary class of graphs is χ-bounded if there exists a function which provides an upper bound for the chromatic number of each graph of the class in terms of the graph's clique number. In this terminology, E. Asplund and B.Grünbaum (1960),  motivated by a question of A. Bielecski (1948), proved that the class of intersection graphs of axis parallel rectangles is χ-bounded, and J. P. Burling, in his Ph.D. thesis (1965), proved that the class of intersection graphs of axis parallel boxes in R^3 is not χ-bounded.

Title: Real matroid Schubert varieties, zonotopes, and virtual Weyl groups

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm.

Abstract: Every linear representation of a matroid determines a matroid Schubert variety whose geometry encodes combinatorics of the matroid. When the representation is over the real numbers, we show that the topology of these varieties is completely determined by the combinatorics of zonotopes. As an application, we compute the fundamental groups. When the real matroid Schubert variety comes from a Coxeter arrangement, we show that the equivariant fundamental group is a “virtual” analogue of the corresponding Weyl group.

Title: A O(log log (rank) ) - competitive algorithm for the matroid secretary problem - Part 2

Abstract: In this talk we continue to review a O(log log(rank) ) - competitive algorithm for the matroid secretary problem (MSP) due to  Feldman, Svensson and Zenklusen, where rank denotes the rank of the given matroid.

Title: A Faster Combinatorial Algorithm for Maximum Bipartite Matching          

Abstract: The maximum bipartite matching problem is among the most fundamental and well-studied problems in combinatorial optimization. A beautiful and celebrated combinatorial algorithm of Hopcroft and Karp (1973) shows that maximum bipartite matching can be solved in $O(m \sqrt{n})$ time on a graph with $n$ vertices and $m$ edges. For the case of very dense graphs, a fast matrix multiplication based approach gives a running time of $O(n^{2.371})$. These results represented the fastest known algorithms for the problem until 2013, when Madry introduced a new approach based on continuous techniques achieving much faster runtime in sparse graphs.

Title: Evolution of random representable matroids

There will be light refreshments starting at 3:00 pm.

Abstract: Inspired by the classical random graph process introduced by Erdos and Renyi in 1960, we introduce two analogous processes for random representable matroids. In the talk we address the following questions, with answers to some already established in existing literature, while others stem from recent research findings:

How does the rank of the random matroid evolve in the process?

When does the first k-circuit appear for each integer k? 

What is the length of the first circuit appearing in the process?

How does the connectivity of the matroid change?

What is the growth rate of the critical number (c.f. chromatic number of graphs) in the process?

When do given (small and large) matroid minors occur in the process?

How about the appearance of submatroids? 

Title: Online Matroid Intersection: Beating Half for Random Arrival

Abstract: We'll take a gander at the paper Online Matroid Intersection: Beating Half for Random Arrival by Guruganesh and Singla. We're given two matroids defined on elements of a common ground set, and elements will arrive one-by-one in a uniformly random order. Whenever an element arrives we'll need to decide to keep them or tell them goodbye forever. The greedy algorithm gives us a competitive ratio of 1/2, but can we do better? Yes! (Slightly, with a simple randomized algorithm). We'll first take a look at the special case of bipartite matching. The results also extend in a natural way to the intersection of k matroids, and non-bipartite matching. If I don't ramble too much, we'll also get to (briefly) discuss a recent improvement for the intersection of 2 matroids.