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Title:Faster linear algebra using treewidth

Abstract:

: We look at the complexity of solving sparse linear systems as a function of the treewidth of the instance. That is, the sparse matrix is associated with a sparse graph, and solutions can be found faster when that graph has low treewidth. We give a parameterized algorithm in system size and treewidth achieving the conjectured optimal performance.

This is joint work with Daniel Grier.

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Title:ÌýAlice, Bob, and colours: An algebraic approach to quantum advantage

Abstract:ÌýTwo-player one-round games exhibit quantum advantage if the availability of quantum resources results in non-classical correlations between the players' answers, which allow outperforming any classical strategy. The first part of the talk illustrates a link between (i) the occurrence of quantum advantage in a game, and (ii) the complexity of the corresponding classical computational problem. In particular, I will show that both (i) and (ii) are determined by the same algebraic invariants of the game, known as polymorphisms. This allows transferring methods from the universal-algebraic theory of constraint satisfaction to the setting of quantum advantage. In particular, this approach yields a complete characterisation of the occurrence of quantum advantage for games played on graphs. ÌýÌý

The second part of the talk outlines recent work on the quantum chromatic number introduced in [Cameron--Montanaro--Newman--Severini--Winter'07]. The gap between this parameter and its classical counterpart is a measure of quantum advantage for the graph colouring game. Using the polymorphic approach, I will show that the maximum gap is linear when the quantum chromatic number makes use of entangled strategies on a constant number of qubits. In contrast, in the case of unlimited entanglement resources, I will prove the existence of graphs whose quantum chromatic number is 3 and whose classical chromatic number is arbitrarily large, conditional to the quantum variants of Khot's d-to-1 Conjecture [Khot'02] and HÃ¥stad's inapproximability of linear equations [HÃ¥stad'01].

Title:ÌýLearning from Coarse Samples

Abstract:Coarsening occurs when the exact value of a sample is not observed.
Instead, only a subset of the sample space containing the exact value is known.
Coarse data naturally arises in diverse fields, including Economics, Engineering, Medical and Biological Sciences, and all areas of the Physical Sciences.
One of the simplest forms of coarsening is rounding, where data values are mapped to the nearest point on a specified lattice.

We survey applications of coarse learning to regression with self-selection bias, regression with second-price auction data, and present details of an SGD-based algorithm for coarse Gaussian mean estimation.

Based on joint work with Alkis Kalavasis and Anay Mehrotra ()

Title:Source characterization of the hypegraphic posets

Abstract: For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic poset $P_\mathbb{H}$ is the transitive closure of the oriented $1$-skeleton of the hypergraphic polytope $\Delta_\mathbb{H}$, which is the Minkowski sum of the standard simplices $\Delta_H$ for each hyperedge $H \in \mathbb{H}$. In 2019, C. Benedetti, N. Bergeron, and J. Machacek established a remarkable correspondence between the transitive closure of the oriented $1$-skeleton of $\Delta_\mathbb{H}$ and the flip graph on acyclic orientations of $\mathbb{H}$. Viewing an orientation of $\mathbb{H}$ as a map $A$ from $\mathbb{H}$ to $[n]$, we define the sources of the acyclic orientations as the values $A(H)$ for each hyperedge $H \in \mathbb{H}$. In a recent paper, N. Bergeron and V.

Pilaud provided a characterization of $P_\mathbb{H}$ based on the sources of acyclic orientations for interval hypergraphs. Specifically, two distinct acyclic orientations $A$ and $B$ of $\mathbb{H}$ are comparable in $P_\mathbb{H}$ if and only if their sources satisfy $A(H) \leq B(H)$ for all hyperedges $H\in \HH$. The goal of this work is to extend this source characterization of $P_\mathbb{H}$ to arbitrary hypergraphs on $[n]$.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm,

Title:Constraining moduli space cohomology by counting graphs

Abstract: In 1992, Kontsevich defined complexes spanned by graphs. TheseÌý
complexes are increasingly prominent in algebraic topology, geometricÌý
group theory and mathematical physics. For instance, a 2021 theorem byÌý
Chan-Galatius and Payne implies that the top-weight cohomology of theÌý
moduli space of curves of genus g is equal to the homology of a specificÌý
graph complex. I will present a new theorem on the asymptotic growthÌý
rate of the Euler characteristic of this graph complex and explain itsÌý
implication on the cohomology of the moduli space of curves. The proofÌý
involves solving a specific graph counting problem.

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Title:ÌýAn O(log log n)-approximate budget-feasible mechanism for subadditive valuations

Abstract:In the setting of budget feasible mechanism design, a buyer wants to purchase items from a set of agents. Each agent holds one item, and incurs a cost of c_i upon supplying the item to the buyer. The buyer wants to maximize the value of the set of items that are bought from the sellers. The buyer has a budget B on the total payments made to the sellers. The cost c_i is private information that the buyer doesn't have access to. The goal is to design a mechanism that is truthful, in the sense that the sellers do not have incentive to deviate from reporting their true costs, and budget feasible, in the sense that the total payments made to the sellers is within some budget B, and that outputs a set whose value is a good approximation to the algorithmic optimum, OPT = max{v(S) : c(S)<=B}.

In this talk, I will present our recent work that obtains an O(log log n)-approximate budget-feasible mechanism when the valuation function is subadditive. This is joint work with Kanstantsin Pashkovich and Chaitanya Swamy.

Title:Splitting algorithms for monotone inclusions with minimalÌýdimension

Abstract: Many situations in convex optimization can be modeled as the problem ofÌýfinding a zero of a monotone operator, which can be regarded as aÌýgeneralization of the gradient of a differentiable convex function. InÌýorder to numerically address this monotone inclusion problem, it isÌývital to be able to exploit the inherent structure of the operatorÌýdefining it. The algorithms in the family of the splitting methodsÌýachieve this by iteratively solving simpler subtasks that are defined byÌýseparately using some parts of the original problem.

In the first part of this talk, we will introduce some of the mostÌýrelevant monotone inclusion problems and present their applications toÌýoptimization. Subsequently, we will draw our attention to a commonÌýanomaly that has persisted in the design of methods in this family: theÌýdimension of the underlying space —which we denote as lifting— of theÌýalgorithms abnormally increases as the problem size grows. This hasÌýdirect implications on the computational performance of the method as aÌýresult of the increase of memory requirements. In this framework, weÌýcharacterize the minimal lifting that can be obtained by splittingÌýalgorithms adept at solving certain general monotone inclusions.ÌýMoreover, we present splitting methods matching these lifting bounds, and thusÌýhaving minimal lifting.

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Title:A Toric Analogue for Greene's Rational Function of a Poset

Abstract: Given a finite poset, Greene introduced a rational function obtained by summing certain rational functions over the linear extensions of the poset. This function has interesting interpretations, and for certain families of posets, it simplifies surprisingly. In particular, Greene evaluated this rational function for strongly planar posets in his work on the Murnaghan–Nakayama formula. Develin, Macauley, and Reiner introduced toric posets, which combinatorially are equivalence classes of posets (or rather acyclic quivers) under the operation of flipping maximum elements into minimum elements and vice versa. In this work, we introduce a toric analogue of Greene's rational function for toric posets, and study its properties. In addition, we use toric posets to show that the Kleiss–Kuijf relations, which appear in scattering amplitudes, are equivalent to a specific instance of Greene's evaluation of his rational function for strongly planar posets. Also in this work, we give an algorithm for finding the set of toric total extensions of a toric poset.

Title:Shifted Parking function

Abstract:Stanley recently introduced the shifted parking function symmetric function as a shifted analogue of the parking function symmetric function and posed the question of what the corresponding combinatorial objects are. This talk will answer that question and explain how the answer connects to projective representations of the symmetric group. Based on joint work with Zach Hamaker.

Title:ÌýA hereditary generalization of Nordhaus-Gaddum graphs

Abstract:ÌýThis talk will be an expanded version of the speaker's CanaDAM talk, the abstract of which is as follows:

Nordhaus and Gaddum proved in 1956 that the sum of the chromatic number of a graph G and its complement is at most |G| + 1. The Nordhaus-Gaddum graphs are the class of graphs satisfying this inequality with equality, and are well-understood. In this paper we consider a hereditary generalization: graphs G for which all induced subgraphs H of G satisfy that the sum of the chromatic numbers of H and its complement are at least |H|. We characterize the forbidden induced subgraphs of this class and find its intersection with a number of common classes, including line graphs. We also discuss chi-boundedness and algorithmic results.