Lecture

Tutte Colloquium - Samuel Jaques

Eva Lee — Tue, 21 Nov 2023 20:32:18 +0000

Tutte Colloquium - Samuel Jaques Eva Lee Tue, 11/21/2023 - 15:32

Title: Wires, bits, and the cost of sorting

Speaker:	Samuel Jaques
Affiliation:	University of ��ݮ��Ƶ
Location:	MC 5501

Abstract: How hard is it to sort a list of n integers? A basic course on algorithms says it's O(n log n) time, but what if the list is enormous - so big you would need to cover the surface of the moon just to store it? At that point we need to worry about how we model the physical architecture of the computer. A reasonable assumption is that, up to constants (possibly very large or small), the different components of a computer -- wires, bits of memory, CPUs, etc. -- should be in roughly constant ratios. Switching to this perspective, I will prove a small theorem: that the physical geometry of a computer implies that the time to sort a list is inversely related to the total length of wires in the computer. This will be a brief window into the science fiction of "galactic algorithms", algorithms at incomprehensible scales that cryptographers analyze to decide whether exponentially hard attacks are truly hard enough.

Tutte Colloquium - Karen Yeats

Eva Lee — Fri, 10 Nov 2023 14:59:55 +0000

Tutte Colloquium - Karen Yeats Eva Lee Fri, 11/10/2023 - 09:59

Title: Diagonal coefficients, graph invariants with the symmetries of Feynman integrals, and the proof of the c_2 completion conjecture

Speaker:	Karen Yeats
Affiliation:	University of ��ݮ��Ƶ
Location:	MC 5501

Abstract: In a scalar field theory the contribution of a Feynman diagram to the beta function of the theory, the Feynman period, can be written as an integral in terms of the (dual) Kirchhoff polynomial of the graph. There are many interesting graph theoretic operations under which this integral is invariant, and we can better understand the period and its geometry by investigating graph invariants with these same symmetries. Recently Erik Panzer found a new such invariant coming from a particular coefficient of the Martin polynomial. Together we used this to prove an over 10 year old conjecture on an arithmetic graph invariant known as the c_2 invariant, and came to understand that diagonal coefficients of Kirchhoff polynomials tie together many of the known graph invariants with the symmetries of Feynman periods and unlock previously inaccessible proofs.

Joint work with Erik Panzer

Tutte Colloquium - Luke Postle

Eva Lee — Fri, 10 Nov 2023 14:40:14 +0000

Tutte Colloquium - Luke Postle Eva Lee Fri, 11/10/2023 - 09:40

Title: Hypergraph Matchings Avoiding Forbidden Submatchings

Speaker:	Luke Postle
Affiliation:	University of ��ݮ��Ƶ
Location:	MC 5501

Abstract: We overview a general theory for finding perfect or almost perfect matchings in a hypergraph G avoiding a given set of forbidden submatchings (which we view as a hypergraph H where V(H)=E(G)). In particular, we have a new common generalization of the classical theorems of Pippenger (for finding an almost perfect matching of G) and Ajtai, Komlos, Pintz, Spencer, and Szemeredi (for finding an independent set in a girth five hypergraph H) into this unified framework. More generally, we proved coloring and list coloring versions, and also generalized this further to when H is a hypergraph with small codegrees. We also discuss applications to various areas of high girth combinatorics. This is joint work with Michelle Delcourt.

Distinguished Tutte Lecture - David B. Shmoys

Eva Lee — Wed, 01 Nov 2023 13:56:02 +0000

Distinguished Tutte Lecture - David B. Shmoys Eva Lee Wed, 11/01/2023 - 09:56

Title: Algorithmic Tools for Congressional Districting: Fairness via Analytics

Speaker:	David B. Shmoys
Affiliation:	Cornell University
Location:	MC 5501

Abstract: The American winner-take-all congressional district system empowers politicians to engineer electoral outcomes by manipulating district boundaries. To date, computational solutions mostly focus on drawing unbiased maps by ignoring political and demographic input, and instead simply optimize for compactness and other related metrics. However, we maintain that this is a flawed approach because compactness and fairness are orthogonal qualities; to achieve a meaningful notion of fairness, one needs to model political and demographic considerations, using historical data.

We will discuss a series of papers that explore and develop this perspective. In the first (joint with Wes Gurnee), we present a scalable approach to explicitly optimize for arbitrary piecewise-linear definitions of fairness; this employs a stochastic hierarchical decomposition approach to produce an exponential number of distinct district plans that can be optimized via a standard set partitioning integer programming formulation. This enables a large-scale ensemble study of congressional districts, providing insights into the range of possible expected outcomes and the implications of this range on potential definitions of fairness. Further work extending this (joint with Julia Allen & Wes Gurnee), shows that many additional real-world constraints can be easily adapted in this framework (such as minimal county splits as was recently required in Alabama legislation in response to the US Supreme Court decision Milligan v. Alabama). In another paper (joint with Nikhil Garg, Wes Gurnee, and David Rothschild), we study the design of multi-member districts (MMDs) in which each district elects multiple representatives, potentially through a non-winner-takes-all voting rule (as was proposed in H.R. 4000). We carry out large-scale analyses for the U.S. House of Representatives under MMDs with different social choice functions, under algorithmically generated maps optimized for either partisan benefit or proportionality. We find that with three-member districts using Single Transferable Vote, fairness-minded independent commissions can achieve proportional outcomes in every state (up to rounding), and this would significantly curtail the power of advantage-seeking partisans to gerrymander.

Tutte Colloquium - Walaa Moursi

Eva Lee — Thu, 19 Oct 2023 19:26:02 +0000

Tutte Colloquium - Walaa Moursi Eva Lee Thu, 10/19/2023 - 15:26

Title: The Chambolle-Pock algorithm revisited: splitting operator and its range with applications

Speaker:	Walaa Moursi
Affiliation:	University of ��ݮ��Ƶ
Location:	MC 5501

Abstract: Primal-dual hybrid gradient (PDHG) is a first-order method for saddle-point problems and convex programming introduced by Chambolle and Pock. Recently, Applegate et al. analyzed the behavior of PDHG when applied to an infeasible or unbounded instance of linear programming, and in particular, showed that PDHG is able to diagnose these conditions. Their analysis hinges on the notion of the infimal displacement vector in the closure of the range of the displacement mapping of the splitting operator that encodes the PDHG algorithm. In this talk, we develop a novel formula for this range using monotone operator theory. The analysis is then specialized to conic programming and further to quadratic programming (QP) and second-order cone programming (SOCP). A consequence of our analysis is that PDHG is able to diagnose infeasible or unbounded instances of QP and of the ellipsoid-separation problem, a subclass of SOCP.

Tutte Colloquium - Sepehr Assadi

Eva Lee — Fri, 13 Oct 2023 17:56:25 +0000

Tutte Colloquium - Sepehr Assadi Eva Lee Fri, 10/13/2023 - 13:56

Title: A Simple Sparsification Algorithm for Maximum Matching with Applications to Graph Streams

Speaker:	Sepehr Assadi
Affiliation:	University of ��ݮ��Ƶ
Location:	MC 5501

Abstract: In this talk, we present a simple algorithm that reduces approximating the maximum weight matching problem in arbitrary graphs to few adaptively chosen instances of the same problem on sparse graphs. In particular, we give an algorithm that, for any eps > 0, computes a (1-eps)-approximate maximum weight matching in arbitrary n-vertex graphs via O(log(n)/eps) call to an algorithm that can solve the same problem on graphs with only O(n/eps) edges.

The algorithm relies on a direct application of the multiplicative weight update method with a self-contained primal-dual analysis. This allows for obtaining a faster convergence rate compared to the applications of generic approaches in this context such as the Plotkin-Shmoys-Tardos framework for packing/covering linear programs.

We will also briefly discuss some applications of this reduction to the design of scalable algorithms on massive graphs, for instance, in the streaming model.

Tutte Colloquium - Nikhil Kumar

Eva Lee — Mon, 25 Sep 2023 12:52:13 +0000

Tutte Colloquium - Nikhil Kumar Eva Lee Mon, 09/25/2023 - 08:52

Title: An Approximate Generalization of the Okamura-Seymour Theorem

Speaker:	Nikhil Kumar
Affiliation:	University of ��ݮ��Ƶ
Location:	MC 5501

Abstract: We consider the problem of multicommodity flows in planar graphs. Okamura and Seymour showed that if all the demands are incident on one face, then the cut-condition is sufficient for routing demands. We consider the following generalization of this setting and prove an approximate max flow-min cut theorem: for every demand edge, there exists a face containing both its end points. We show that the cut-condition is sufficient for routing Ω(1) -fraction of all the demands. To prove this, we give a L1-embedding of the planar metric which approximately preserves distance between all pair of points on the same face.

Tutte Colloquium - Vida Dujmovic

Eva Lee — Fri, 15 Sep 2023 14:44:49 +0000

Tutte Colloquium - Vida Dujmovic Eva Lee Fri, 09/15/2023 - 10:44

Title: Proof of the Clustered Hadwiger Conjecture

Speaker:	Vida Dujmovic
Affiliation:	University of Ottawa
Location:	MC 5501

Abstract: Hadwiger's Conjecture asserts that every Kh-minor-free graph is properly (h-1)-colourable. We prove the following improper analogue of Hadwiger's Conjecture: for fixed h, every Kh-minor-free graph is (h-1)-colourable with monochromatic components of bounded size.The number of colours is best possible regardless of the size of monochromatic components. It solves an open problem of Edwards, Kang, Kim, Oum and Seymour [SIAM J. Disc. Math. 2015], and concludes a line of research initiated in 2007. Similarly, for fixed t ⩾ s, we show that every Ks,t-minor-free graph is (s+1)-colourable with monochromatic components of bounded size. The number of colours is best possible, solving an open problem of van de Heuvel and Wood [J. London Math. Soc. 2018]. We actually prove a single theorem from which both of the above results are immediate corollaries.

This is joint work with Louis Esperet, Pat Morin and David R. Wood.

Tutte Colloquium - Bill Jackson

Eva Lee — Thu, 27 Jul 2023 21:08:00 +0000

Tutte Colloquium - Bill Jackson Eva Lee Thu, 07/27/2023 - 17:08

Title: Rigidity of Simplicial Complexes

Speaker:	Bill Jackson
Affiliation:	Queen Mary University of London
Location:	MC 5501

Abstract: A simplicial k-cycle is an abstract simplicial k-complex in which every (k-1)-face belongs to an even number of k-faces. A simplicial k-circuit is a minimal simplicial k-cycle (in the sense that none of its proper subcomplexes are simplicial k-cycles). The set of all simplicial k-circuits with vertex set V is the set of circuits of the simplicial k-matroid whose groundset is the set of all (k+1)-subsets of V. Examples of simplicial k-circuits occur naturally as boundary complexes of (k+1)-dimensional simplicial polytopes or, more generally, connected simplicial k-dimensional manifolds. In particular, every connected triangulated surface is a simplicial 2-circuit.

A celebrated result of Cauchy implies that the 1-skeleton of every simplicial convex polyhedron is rigid. This was extended to simplicial convex d-dimensional polytopes by Whiteley in 1984. These results become false if we remove the convexity constraint but Fogelsanger showed in his PhD thesis in 1988 that they continue to hold in the generic case. More precisely, he showed that the 1-skeleton of every simplicial (d-1)-circuit is generically rigid in \(\mathbb{R}^d\) whenever d is greater or equal than 3. Together with James Cruickshank and Shin-ichi Tanigawa, we recently adapted Fogelsanger's proof technique to obtain results on the global rigidity of simplicial (d-1)-circuits and the rigidity of symmetric simplicial (d-1)-circuits in \(\mathbb{R}^d\).

I will give a brief introduction to the theory of rigid frameworks, then describe Fogelsanger’s ingenious proof technique and outline how it can be adapted to work for global and symmetric rigidity.

Joint Tutte Colloquium & Algorithms and Complexity Seminar - Leonid Gurvits

Eva Lee — Wed, 12 Jul 2023 15:50:31 +0000

Joint Tutte Colloquium & Algorithms and Complexity Seminar - Leonid Gurvits Eva Lee Wed, 07/12/2023 - 11:50

Title: Combinatorial and complexity theoretic aspects of Stabilities and Controllabilities of linear switched systems(discrete and continuous time)

Speaker:	Leonid Gurvits
Affiliation:	The City College of New York
Location:	MC 5501

Abstract: I will talk about my "pre-hyperbolic" research, some of it done jointly with Alex Samorodnitsky and Alex Olshevsky.

For example:

1. One of the results I plan to present gives a criteria for the existence of n×n real Hadamard Matrix expressed in terms of best approximation of so called joint spectral radius by quadratic Lyapunov Functions.

2. Another one is about a complexity theoretic separation in analysis of Switched Absolute Stability for Open Classical(easy) and Quantum(hard) Linear Continuous Time Systems.

3. Time permitting, a separation between separable vs entangled states will be presented in terms regular vs nonregular languages.

The talk should be accessible to wide mathematical audience as all necessary notions will be introduced/explained on the fly.