Events - 2022

Title: Generating k EPR-pairs from an n-party resource state

Abstract: Motivated by quantum network applications over classical channels, we initiate the study of n-party resource states from which LOCC protocols can create EPR-pairs between any k disjoint pairs of parties.

Title: Cameron-Liebler Sets in Permutation Groups

Abstract: Let $G \leq S_{n}$ be a transitive permutation group. Given $i,j \in [n]$, by $x_{i\to j}$, denote the characteristic function of the set $\{g \in G\ :\ g(i)=j\}$. A Cameron-Liebler set (CL set) in $G$ is a set which is represented by a Boolean function in the linear span of $\{x_{i\to j} \ :\ (i,j)\in [n]^2\}$. These are analogous to Boolean degree 1 functions on the hypercube and to Cameron-Liebler line classes in $PG(3,q)$. Sets of the form $\{g\ : g(i)\in X\}$ and $\{g\ : \ i \in g(X)\}$ (for $i \in [n]$ and $X \subset [n]$) are canonically occurring examples of CL sets. A result of Ellis et.al, shows that all CL sets in the $S_{n}$ are canonnical. In this talk, we will demonstrate many examples with ``exotic'' CL sets. Of special interest is an exotic CL set in $PSL(2,q)$ (with $q \equiv 3 \pmod{4}$), a 2-transitive group, just like $S_{n}$. The talk is based on ongoing joint work with Jozefien D'haeseleer and Karen Meagher.

Title: Approximation algorithm for stochastic k-TSP

Abstract: The input of the deterministic k-TSP problem consists of a metric complete graph with root p in which the nodes are assigned a fixed non-negative reward. The objective is to construct a p-rooted path of minimum length that collects total reward at least k. In this talk we will explore a stochastic variant of this problem in which the rewards assigned to the nodes are independent random variables, and the objective is to derive a policy that minimizes the expected length of a p-rooted path that collects total reward at least k. We will discuss approximation algorithms for this problem proposed in a paper by Ene, Nagarajan and Saket, and a paper by Jiang, Li, Liu and Singla.

Title: Algebraicity of solutions of functional equations with one catalytic variable

Abstract: Abstract: Numerous combinatorial enumeration problems reduce to the study of functional equations which can be solved by a uniform method introduced by Bousquet-Mélou and Jehanne in 2006. In my talk, I will first briefly explain this result and its proof. Then I will present a new generalization of it to the case of systems of functional equations with one catalytic variable. The method is constructive and yields an algorithm for computing the minimal polynomials of interest.

Title: A recursive spectral bound for independence

Abstract: We discuss an upper bound for the independence number of a graph, in the spirit of the well-known Hoffman bound. Our bound involves the largest Laplacian eigenvalue of the graph; more surprisingly, it also involves the independence number of a certain induced graph. We illustrate the bound on several examples.

Title: Greedy algorithm for stochastic matching is a 2-approximatio

Abstract: We will discuss the greedy algorithm for the stochastic matching problem. In this problem, we are given an undirected graph where each edge is assigned a probability p_e in [0, 1] and each vertex is assigned a patience t_v in Z+. We begin each step by probing an edge e which is not adjacent to any edges in our matching. The probe will succeed with probability p_e, and if it does, we add e to our matching. Otherwise, we may not probe e again. We also may not probe edges adjacent to a vertex v more than t_v times. The goal is to maximize the number of edges we add to our matching. 

Title: Exact Zarankiewicz numbers through linear hypergraphs

Abstract: The \emph{Zarankiewicz number} $Z_{2,2}(m,n)$ is usually defined as the maximum number of edges in a bipartite graph with parts of sizes $m$ and $n$ that has no $K_{2,2}$ subgraph. An equivalent definition is that $Z_{2,2}(m,n)$ is the greatest total degree of a linear hypergraph with $m$ vertices and $n$ edges. A hypergraph is \emph{linear} if each pair of vertices appear together in at most one edge. The equivalence of the two definitions can be seen by considering the bipartite incidence graph of the linear hypergraph.

Title: Approximation algorithms for stochastic orienteering  

Abstract: This week we revisit the stochastic orienteering problem in which we are given a metric graph where each node has a deterministic reward and a random size. The goal is to adaptively decide on which nodes to visit to maximize the expected reward, while not exceeding the budget B on the distance plus the size of jobs processed.

Title: The Hat Guessing Number of Graphs

Abstract:  The hat guessing number HG(G) of a graph G on n vertices is defined in terms of the following game: n players are placed on the n vertices of G, each wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number HG(G) is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.

Title: Forbidding some induced cycles in a graph

Abstract:  We call an induced cycle of length at least four a hole. The parity of a hole is the parity of its length. Forbidding holes of certain lengths in a graph has deep structural implications. In 2006, Chudnovksy, Seymour, Robertson, and Thomas famously proved that a graph is perfect if and only if it does not contain an odd hole or a complement of an odd hole. In 2002, Conforti, Cornuéjols, Kapoor, and Vušković provided a structural description of the class of even-hole-free graphs.

Title: Recursions and Proofs in Coxeter-Catalan combinatorics

Abstract:  The collection of parking functions under a natural Sn-action (which has Catalan-many orbits) has been a central object in Algebraic Combinatorics since the work of Haiman more than 30 years ago. One of the lines of research spawned around it was towards defining and studying analogous objects for real and complex reflection groups W; the main candidates are known as the W-non-nesting and W-non-crossing parking functions.

Title: Sylvester, Gallai, and their complex relatives

Abstract: Given any finite set of points in the real plane, not all collinear, there is a line in the plane that contains exactly two of them. This pretty result was conjectured by Sylvester in 1893 and proved by Gallai in 1944. We will present an extension of the result to higher dimensional complex spaces and discuss some related conjectures. This is joint work with Matthew Kroeker.