Events

A strongly polynomial algorithm for linear programs with at most two non-zero entries per row or column.

Abstract:We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Primal and dual feasibility were shown by Megiddo (SICOMP '83) and Végh (MOR '17) respectively. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale's 9th problem. Our approach is based on the recent primal-dual interior point method (IPM) due to Allamigeon, Dadush, Loho, Natura and Végh (FOCS '22). The number of iterations needed by the IPM is bounded, up to a polynomial factor in the number of inequalities, by the straight line complexity of the central path. Roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path. As our main contribution, we show that the straight line complexity of any minimum cost generalized flow instance is polynomial in the number of arcs and vertices. By applying a reduction of Hochbaum (ORL '04), the same bound applies to any linear program with at most two non-zeros per column or per row. To be able to run the IPM, one requires a suitable initial point. For this purpose, we develop a novel multistage approach, where each stage can be solved in strongly polynomial time given the result of the previous stage. Beyond this, substantial work is needed to ensure that the bit complexity of each iterate remains bounded during the execution of the algorithm. For this purpose, we show that one can maintain a representation of the iterates as a low complexity convex combination of vertices. Our approach is black-box and can be applied to any log-barrier path following method. 

Bento Natura is an Assistant Professor in Industrial Engineering and Operations Research (IEOR) at Columbia University. He spent two years as a Postdoctoral Fellow at Georgia Tech, Brown University, and UC Berkeley. Prior to that, he obtained his PhD in Mathematics from the London School of Economics.

His research interests are focused on the areas of algorithms, optimization, and game theory, with a special emphasis on the theory of linear programming.

Title: Diophantine tuples, graphs, and additive combinatorics

Abstract: In this talk, we investigate the multiplicative structure of a shifted multiplicative subgroup and its connections with additive combinatorics and the theory of Diophantine tuples and Diophantine graphs. First, we show that if a nontrivial shift of a multiplicative subgroup $G$ contains a product set $AB$, then $|A||B|$ is essentially bounded by $|G|$, refining a well-known consequence of a classical result by Vinogradov. Second, we provide a sharper upper bound of $M_k(n)$, or the clique number of Diophantine graphs, which is the largest size of a set such that each pairwise product of its elements is $n$ less than a $k$-th power, refining the recent result of Dixit, Kim, and Murty.  One main ingredient in our proof is the first non-trivial upper bound on the maximum size of a generalized Diophantine tuple over a finite field. In addition, we determine the maximum size of an infinite family of generalized Diophantine tuples over finite fields with square order, which is of independent interest.  

We also present a significant progress towards a conjecture of S\'{a}rk\"{o}zy on the multiplicative decompositions of shifted multiplicative subgroups. In particular, we prove that for almost all primes $p$, the set $\{x^2-1: x \in \F_p^*\} \setminus \{0\}$ cannot be decomposed as the product of two sets in $\F_p$ non-trivially.

Title: Non-measurable colourings avoiding large distances

Abstract: "In 1983, F{\"u}rstenberg, Katznelson, and Weiss proved that for every finite measurable colouring of the plane, there exists a $d_0$ such that for every $d \ge d_0$ there is a monochromatic pair of points at distance $d$. In contrast to this, we show that there is a finite colouring avoiding arbitrarily large distances.

Joint work with Rutger Campbell."

Approximately Counting Flows via Generating Function Optimization

Abstract: In this talk, we will present recent new lower bounds on the number of non-negative integer flows on a directed acyclic graph with specified total vertex flows (or equivalently, the number of lattice points of a given flow polytope, or the coefficients of the A-type Kostant partition function). We will also give a sketch of the proof, which involves three main parts: (1) prove a certain log-concavity property of the associated multivariate generating function, (2) prove bounds on the coefficients in terms of an associated optimization problem, and (3) dualize the optimization problem to obtain the desired lower bounds. If time permits, we will also briefly discuss other applications of this technique, including to approximating Kostka numbers and to the traveling salesperson problem. Joint work with Alejandro Morales, and with Petter Brändén and Igor Pak.

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

Title: LP based approximation algorithm for an stochastic matching problem

Abstract: Consider the random graph model where each edge e has a fixed weight w_e and it is independently present in the graph with probability p_e​. Given these probabilities, we want to construct a maximum weight matching in the graph. One can only determine if an edge is present by querying it, and if an edge is present, it must be irrevocably included in the matching. Additionally, each vertex i can be queried no more than t_i times. The goal is to device an adaptive policy (algorithm) to query the edges of the graph one by one in order to maximize the expected weight of the matching.

In this talk we present an elegant LP-based constant-factor approximation algorithm with respect to the optimal adaptive policy for the problem.

This is one of the results due to Bansal, Gupta, Li, Mestre, Nagarajan, and Rudra, in their paper "When LP is the Cure for your Matching Woes" from 2011.

Graph Property Testing using the Container Method

Abstract: The Graph and Hypergraph Container Methods have recently been used to obtain multiple striking results across different areas of mathematics. In this talk, we will see how the graph container method is particularly well-suited for the study of some fundamental problems in graph property testing.

The main problem we will discuss in the talk is the Independent Set Testing problem introduced by Goldreich, Goldwasser, and Ron (1998). In this problem, we are given oracle access to a graph on $n$ vertices that either (i) contains an independent set on $\rho n$ vertices, or (ii) is $\epsilon$-far from the property in the sense that at least $\epsilon n^2$ edges must be removed from the graph to make it have an independent set of this size. We will introduce a new container lemma for the latter class of graphs and we will show how this lemma can be used to obtain a near-optimal solution to the Independent Set Testing problem. We will also discuss how variants and extensions of the new container lemma can be used to prove a variety of other results in property testing.

This is joint work with Cameron Seth.

Title: Combinatorial Nullstellensatz and the Erdős box problem

Abstract: In the talk, I will show how Lasoń’s generalization of Alon’s Combinatorial Nullstellensatz can be used to obtain lower bounds on Turán numbers of complete r-partite r-uniform hypergraphs. As an example, I will give a short and simple explicit construction of a hypergraph free of copies of the complete r-partite r-uniform  hypergraph with parts of size 2, thereby providing a lower bound for the so-called Erdős box problem. This asymptotically matches best known bounds when r ≤ 4. 

Title:Coefficient Positivity and Analytic Combinatorics

 Abstract: Given a rational function analytic at the origin, are its power series coefficients positive? We first motivate this question with historical background, examples and connections to decidability questions in combinatorics. We then present a method for proving positivity for a certain class of multivariate rational functions, by constructing explicit error bounds for an asymptotic expansion using analytic combinatorics in several variables. After applying our method to a couple of examples in the literature, including re-proving a (recently proved) conjecture of Gillis, Reznick and Zeilberger from 1983, we discuss the feasibility of completing certain steps in our analysis using methods from computer algebra.

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

Abstract: In this talk, we will cover the paper of Svensson and Tarnawski that shows that perfect matching in general (non-bipartite) graphs in is quasi-NC. Similar to the work of Fenner, Gurjar and Thierauf (covered earlier in the reading group), the approach is to derandomize the isolation lemma for the perfect matching polytope by applying weight functions to iteratively restrict to subfaces of the polytope. However, the perfect matching polytope in general graphs is not as well-structured as it is in bipartite graphs. The faces of the polytope no longer correspond to subgraphs and now involve additional tight odd set constraints that need to be dealt with. This makes it so that a cycle with non-zero circulation may still exist in the support of the new face. Additionally, the existence of odd cycles in the graph breaks the cycle counting argument used in the paper of Fenner, Gurjar, Thierauf. We will see how Svensson and Tarnawski deal with these issues in the talk.

Title: Purifying arbitrarily noisy quantum states

Abstract: Quantum state purification is the task of recovering a nearly pure copy of an unknown pure quantum state using multiple noisy copies of the state. We derive an efficient purification procedure based on the swap test for qudits of any dimension, starting with any initial error parameter. For constant initial error parameter and dimension, our procedure has sample complexity asymptotically optimal in the final error parameter, and almost matches the known optimal protocol for qubits. Our protocol has a simple recursive structure that can be applied when the states are provided one at a time in a streaming fashion, requiring only a small quantum memory to implement.  Joint work with Andrew Childs, Honghao Fu, Zhi Li, Maris Ozols, Vedang Vyas.