Future students

Title:The first-order logic of graphs

Abstract:Over the last ten years, many wonderful connections have been established between structural graph theory, computational complexity, and finite model theory. We give an overview of this area, focusing on recent progress towards understanding the "stable" case. We do not assume any familiarity with first-order logic

Title: Refuting semirandom CSPs via spectral graph theory techniques

Abstract:

: In this talk, we will consider recent spectral techniques, developed by [HKM23] and [GKM22], for combinatorial and algorithmic problems. We shall focus, in particular, on designing algorithms for refuting semirandom instances of constraint satisfaction problems. The main component of the talk is a reduction to studying spectral properties of so-called "Kikuchi graphs" corresponding to a system of homogeneous degree-q multilinear polynomials.

No prerequisites in spectral graph theory beyond basic linear algebra are assumed.

[HKM23]: A simple and sharper proof of the hypergraph Moore bound

[GKM22]: Algorithms and Certificates for Boolean CSP Refutation: "Smoothed is no harder than Random"

Title:Power sum expansions for the Kromatic symmetric function

Abstract:The Kromatic symmetric function was introduced by Crew, Pechenik, and Spirkl (2023) as a K-analogue of Stanley's chromatic symmetric function. While the chromatic symmetric function encodes proper colorings of a graph (where each vertex gets a color and adjacent vertices get different colors), the Kromatic symmetric function encodes proper set colorings (where each vertex gets a nonempty set of colors and adjacent vertices get non-overlapping color sets). The expansion of the chromatic symmetric function in the basis of power sum symmetric functions has several nice interpretations, including one in terms of source components of acyclic orientations, due to Bernardi and Nadeau (2020). We lift that expansion formula to give expansion formulas for the Kromatic symmetric function using a few different K-analogues of the power sum basis. Our expansions are based on Lyndon heaps, introduced by Lalonde (1995), which are representatives for certain equivalence classes of acyclic orientations on clan graphs (graphs formed from the original graph by removing vertices and adding extra copies of vertices).

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

Title: A Sudoku Baranyai's Theorem

Abstract: Motivated by constructing higher dimensional Sudokus we generalize the famous Baranyai's theorem. Let$n=\prod_{i=1}^d a_i$. Suppose that an $n\times\dots \times n$ ($d$ times) array $L$ is partitioned into$n/a_1\times\dots\times n/a_d$ sub‐arrays (called blocks). Can we color the $n^d$ cells of $L$ with $n^{d21}$ colors so that each layer (obtained by fixing one coordinate) and each $n/a_1\times\dots\times n/a_d$block contains each color exactly once? We generalize the well‐know theorem of Baranyai to answer this queestion. The case $d=2 a_1=a_2=3$ corresponds to the usual Sudoku. We also provide finite fieldconstruction of various related objects. This is joint work with Sho Suda.

Title:Unsplittable Multicommodity Flows in Outerplanar Graphs

Abstract:

Given a graph G with edge capacities and multiple source-sink pairs, each with an associated demand, the multicommodity flow problem consists of routing all demands simultaneously without violating edge capacities. The graph obtained by including an edge (s_i,t_i) for a demand with source-sink s_i,t_i is called the demand graph H.

A multicommodity flow is called unsplittable if all the flow between a source-sink pair is routed along a single path. In general, existence of a feasible (fractional) flow does not imply the existence of an unsplittable flow, even in very simple settings.  This leads to a natural question: given a feasible flow, does there exist an unsplittable flow which satisfies all the demands and violates the edge capacities (in an additive sense) by at most a small factor times the value of the maximum demand Dmax? 

Dinitz, Garg and Goemans proved that in the single-source setting (i.e., when H is a star), any feasible fractional flow can be converted into an unsplittable flow that violates the edge capacities by no more than Dmax. 

The problem is significantly less understood when the demand graph H is arbitrary. Schrijver, Seymour and Winkler proved that if G is a cycle, then any feasible multicommodity flow can be converted into an unsplittable one that violates the edge capacities by at most 3Dmax/2. Before our work, cycles were the only known nontrivial class of graphs for which an unsplittable flow was guaranteed to exist, incurring at most an additive O(Dmax) violation of edge capacities, whenever a feasible flow existed. In this talk, I will discuss how to extend this result to the class of outerplanar graphs.

This is a joint work with Nikhil Kumar.

Title: A hereditary generalization of Nordhaus-Gaddum graphs

Abstract: This 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.

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: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:Regularity lemmas for hypergraphs of bounded VC dimension: improved bounds

Abstract:An important result at the interface of graph theory and logic is that graphs of bounded VC dimension have (small) homogeneous vertex-partitions, i.e., partitions where almost every pair of parts has density close to 0 or 1. Recently, Chernikov and Towsner proved a hypergraph generalization of this fact. The quantitative aspects of their result remain open. I will present some recent progress on this problem, answering two questions of Terry. This is a joint work with Asaf Shapira and Yuval Wigderson.

Title: An 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.