# Institute for Computational and Experimental Research in Mathematics (ICERM)

January 30, 2023 - February 3, 2023
##### Monday, January 30, 2023
• 9:30 - 9:50 am EST
Check In
11th Floor Collaborative Space
• 9:50 - 10:00 am EST
Welcome
11th Floor Lecture Hall
• Brendan Hassett, ICERM/Brown University
• 10:00 - 11:00 am EST
Matching Theory and School Choice
Seminar - 11th Floor Lecture Hall
• Speaker
• Yuri Faenza, Columbia University
• Session Chair
• Jon Lee, University of Michigan
###### Abstract
Many questions in resource allocation can be formulated as matching problems, where nodes represent the agents/goods, and each node corresponding to an agent is endowed with a preference profile on the (sets of) its neighbors in the graph. Starting with the classical marriage setting by Gale and Shapley, we will investigate algorithmic and structural properties of these models, and discuss applications to the problem of allocating seats in public schools.
• 11:00 - 11:30 am EST
Coffee Break
11th Floor Collaborative Space
• 11:30 am - 12:30 pm EST
Matching Theory and School Choice
Seminar - 11th Floor Lecture Hall
• Speaker
• Yuri Faenza, Columbia University
• Session Chair
• Jon Lee, University of Michigan
###### Abstract
Many questions in resource allocation can be formulated as matching problems, where nodes represent the agents/goods, and each node corresponding to an agent is endowed with a preference profile on the (sets of) its neighbors in the graph. Starting with the classical marriage setting by Gale and Shapley, we will investigate algorithmic and structural properties of these models, and discuss applications to the problem of allocating seats in public schools.
• 12:30 - 2:30 pm EST
Lunch/Free Time
• 2:30 - 3:30 pm EST
Binary polynomial optimization: theory, algorithms, and applications
Seminar - 11th Floor Lecture Hall
• Speaker
• Session Chair
• Marcia Fampa, Federal University of Rio de Janeiro
###### Abstract
In this mini-course, I present an overview of some recent advances in the theory of binary polynomial optimization together with specific applications in data science and machine learning. First utilizing a hypergraph representation scheme, I describe the connection between hypergraph acyclicity and the complexity of unconstrained binary polynomial optimization. As a byproduct, I present strong linear programming relaxations for general binary polynomial optimization problems and demonstrate their impact via extensive numerical experiments. Finally, I focus on two applications from data science, namely, Boolean tensor factorization and higher-order Markov random fields, and demonstrate how our theoretical findings enable us to obtain efficient algorithms with theoretical performance guarantees for these applications.
• 3:30 - 4:00 pm EST
Coffee Break
11th Floor Collaborative Space
• 4:00 - 5:00 pm EST
Binary polynomial optimization: theory, algorithms, and applications
Seminar - 11th Floor Lecture Hall
• Speaker
• Session Chair
• Marcia Fampa, Federal University of Rio de Janeiro
###### Abstract
In this mini-course, I present an overview of some recent advances in the theory of binary polynomial optimization together with specific applications in data science and machine learning. First utilizing a hypergraph representation scheme, I describe the connection between hypergraph acyclicity and the complexity of unconstrained binary polynomial optimization. As a byproduct, I present strong linear programming relaxations for general binary polynomial optimization problems and demonstrate their impact via extensive numerical experiments. Finally, I focus on two applications from data science, namely, Boolean tensor factorization and higher-order Markov random fields, and demonstrate how our theoretical findings enable us to obtain efficient algorithms with theoretical performance guarantees for these applications.
• 5:00 - 6:30 pm EST
Reception
11th Floor Collaborative Space
##### Tuesday, January 31, 2023
• 9:00 - 10:00 am EST
Approximation Algorithms for Network Design Problems
Seminar - 11th Floor Lecture Hall
• Speaker
• Vera Traub, University of Bonn
• Session Chair
• Laura Sanità, Bocconi University of Milan
###### Abstract
The goal of network design is to construct cheap networks that satisfy certain connectivity requirements. A celebrated result by Jain [Combinatorica, 2001] provides a 2-approximation algorithm for a wide class of these problems. However, even for many very basic special cases nothing better is known. In this lecture series, we present an introduction and some of the new techniques underlying recent advances in this area. These techniques led for example to a new algorithm for the Steiner Tree Problem and to the first better-than-2 approximation algorithm for Weighted Connectivity Augmentation.
• 10:00 - 10:30 am EST
Coffee Break
11th Floor Collaborative Space
• 10:30 - 11:30 am EST
Approximation Algorithms for Network Design Problems
Seminar - 11th Floor Lecture Hall
• Speaker
• Vera Traub, University of Bonn
• Session Chair
• Laura Sanità, Bocconi University of Milan
###### Abstract
The goal of network design is to construct cheap networks that satisfy certain connectivity requirements. A celebrated result by Jain [Combinatorica, 2001] provides a 2-approximation algorithm for a wide class of these problems. However, even for many very basic special cases nothing better is known. In this lecture series, we present an introduction and some of the new techniques underlying recent advances in this area. These techniques led for example to a new algorithm for the Steiner Tree Problem and to the first better-than-2 approximation algorithm for Weighted Connectivity Augmentation.
• 11:45 am - 1:00 pm EST
Problem Session
11th Floor Lecture Hall
• 1:00 - 3:00 pm EST
Lunch/Free Time
• 3:00 - 5:00 pm EST
Poster Session / Coffee Break
Poster Session - 11th Floor Collaborative Space
##### Wednesday, February 1, 2023
• 9:00 - 10:00 am EST
Polynomial optimization on finite sets
Seminar - 11th Floor Lecture Hall
• Speaker
• Mauricio Velasco, Universidad de Los Andes
• Session Chair
• Jesús De Loera, University of California, Davis
###### Abstract
If $X\subseteq \mathbb{R}^n$ is a finite set then every function on $X$ can be written as the restriction of a polynomial in n-variables. As a result, polynomial optimization on finite sets is literally the same as general (nonlinear) optimization on such sets. Thinking of functions as polynomials, however, provides us with plenty of additional structures which can be leveraged for constructing better (or at least different) optimization algorithms. In these lectures, we will overview some of the key problems and results coming from this algebraic point of view. Specifically, we will discuss: How to prove that a polynomial function is nonnegative on a finite set X? What kind of algebraic certificates (proofs) are available and what can we say about their size and complexity? If the set $X$ has symmetries, can we leverage them in some systematic way that is useful for optimization? Characterizing the affine linear functions that are nonnegative on $X$ gives a description of the polytope $P={\rm Conv}(X)$. Stratifying such functions by the degree of their nonnegativity certificates leads to (semidefinite) hierarchies of approximation for the polytope $P$ and it is natural to ask about their speed of convergence and its relationship with the combinatorics of $P$ Finally, if time permits we will discuss some recent ideas combining the above methods with reinforcement learning as a way to improve scalability for combinatorial optimization problems. The results in (1),(2), (3) above are due to Blekherman, Gouveia, Laurent, Nie, Parrilo, Saunderson, Thomas, and others. These lectures intend to be a self-contained introduction to this vibrant and exciting research area.
• 10:00 - 10:30 am EST
Coffee Break
11th Floor Collaborative Space
• 10:30 - 11:30 am EST
Polynomial optimization on finite sets
Seminar - 11th Floor Lecture Hall
• Speaker
• Mauricio Velasco, Universidad de Los Andes
• Session Chair
• Jesús De Loera, University of California, Davis
###### Abstract
If $X\subseteq \mathbb{R}^n$ is a finite set then every function on $X$ can be written as the restriction of a polynomial in n-variables. As a result, polynomial optimization on finite sets is literally the same as general (nonlinear) optimization on such sets. Thinking of functions as polynomials, however, provides us with plenty of additional structures which can be leveraged for constructing better (or at least different) optimization algorithms. In these lectures, we will overview some of the key problems and results coming from this algebraic point of view. Specifically, we will discuss: How to prove that a polynomial function is nonnegative on a finite set X? What kind of algebraic certificates (proofs) are available and what can we say about their size and complexity? If the set $X$ has symmetries, can we leverage them in some systematic way that is useful for optimization? Characterizing the affine linear functions that are nonnegative on $X$ gives a description of the polytope $P={\rm Conv}(X)$. Stratifying such functions by the degree of their nonnegativity certificates leads to (semidefinite) hierarchies of approximation for the polytope $P$ and it is natural to ask about their speed of convergence and its relationship with the combinatorics of $P$ Finally, if time permits we will discuss some recent ideas combining the above methods with reinforcement learning as a way to improve scalability for combinatorial optimization problems. The results in (1),(2), (3) above are due to Blekherman, Gouveia, Laurent, Nie, Parrilo, Saunderson, Thomas, and others. These lectures intend to be a self-contained introduction to this vibrant and exciting research area.
• 11:40 - 11:45 am EST
Group Photo
11th Floor Lecture Hall
• 11:45 am - 2:00 pm EST
Lunch/Free Time
• 2:00 - 3:00 pm EST
Matching Theory and School Choice
Seminar - 11th Floor Lecture Hall
• Speaker
• Yuri Faenza, Columbia University
• Session Chair
• Jon Lee, University of Michigan
###### Abstract
Many questions in resource allocation can be formulated as matching problems, where nodes represent the agents/goods, and each node corresponding to an agent is endowed with a preference profile on the (sets of) its neighbors in the graph. Starting with the classical marriage setting by Gale and Shapley, we will investigate algorithmic and structural properties of these models, and discuss applications to the problem of allocating seats in public schools.
• 3:00 - 3:30 pm EST
Coffee Break
11th Floor Collaborative Space
• 3:30 - 4:30 pm EST
Matching Theory and School Choice
Seminar - 11th Floor Lecture Hall
• Speaker
• Yuri Faenza, Columbia University
• Session Chair
• Jon Lee, University of Michigan
###### Abstract
Many questions in resource allocation can be formulated as matching problems, where nodes represent the agents/goods, and each node corresponding to an agent is endowed with a preference profile on the (sets of) its neighbors in the graph. Starting with the classical marriage setting by Gale and Shapley, we will investigate algorithmic and structural properties of these models, and discuss applications to the problem of allocating seats in public schools.
##### Thursday, February 2, 2023
• 9:00 - 10:00 am EST
Approximation Algorithms for Network Design Problems
Seminar - 11th Floor Lecture Hall
• Speaker
• Vera Traub, University of Bonn
• Session Chair
• Laura Sanità, Bocconi University of Milan
###### Abstract
The goal of network design is to construct cheap networks that satisfy certain connectivity requirements. A celebrated result by Jain [Combinatorica, 2001] provides a 2-approximation algorithm for a wide class of these problems. However, even for many very basic special cases nothing better is known. In this lecture series, we present an introduction and some of the new techniques underlying recent advances in this area. These techniques led for example to a new algorithm for the Steiner Tree Problem and to the first better-than-2 approximation algorithm for Weighted Connectivity Augmentation.
• 10:00 - 10:30 am EST
Coffee Break
11th Floor Collaborative Space
• 10:30 - 11:30 am EST
Approximation Algorithms for Network Design Problems
Seminar - 11th Floor Lecture Hall
• Speaker
• Vera Traub, University of Bonn
• Session Chair
• Laura Sanità, Bocconi University of Milan
###### Abstract
The goal of network design is to construct cheap networks that satisfy certain connectivity requirements. A celebrated result by Jain [Combinatorica, 2001] provides a 2-approximation algorithm for a wide class of these problems. However, even for many very basic special cases nothing better is known. In this lecture series, we present an introduction and some of the new techniques underlying recent advances in this area. These techniques led for example to a new algorithm for the Steiner Tree Problem and to the first better-than-2 approximation algorithm for Weighted Connectivity Augmentation.
• 11:30 am - 1:30 pm EST
Lunch/Free Time
• 1:30 - 2:30 pm EST
Binary polynomial optimization: theory, algorithms, and applications
Seminar - 11th Floor Lecture Hall
• Speaker
• Session Chair
• Marcia Fampa, Federal University of Rio de Janeiro
###### Abstract
In this mini-course, I present an overview of some recent advances in the theory of binary polynomial optimization together with specific applications in data science and machine learning. First utilizing a hypergraph representation scheme, I describe the connection between hypergraph acyclicity and the complexity of unconstrained binary polynomial optimization. As a byproduct, I present strong linear programming relaxations for general binary polynomial optimization problems and demonstrate their impact via extensive numerical experiments. Finally, I focus on two applications from data science, namely, Boolean tensor factorization and higher-order Markov random fields, and demonstrate how our theoretical findings enable us to obtain efficient algorithms with theoretical performance guarantees for these applications.
• 2:30 - 3:30 pm EST
Coffee Break
11th Floor Collaborative Space
• 3:30 - 4:30 pm EST
Binary polynomial optimization: theory, algorithms, and applications
Seminar - 11th Floor Lecture Hall
• Speaker
• Session Chair
• Marcia Fampa, Federal University of Rio de Janeiro
###### Abstract
In this mini-course, I present an overview of some recent advances in the theory of binary polynomial optimization together with specific applications in data science and machine learning. First utilizing a hypergraph representation scheme, I describe the connection between hypergraph acyclicity and the complexity of unconstrained binary polynomial optimization. As a byproduct, I present strong linear programming relaxations for general binary polynomial optimization problems and demonstrate their impact via extensive numerical experiments. Finally, I focus on two applications from data science, namely, Boolean tensor factorization and higher-order Markov random fields, and demonstrate how our theoretical findings enable us to obtain efficient algorithms with theoretical performance guarantees for these applications.
##### Friday, February 3, 2023
• 10:00 - 11:00 am EST
Polynomial optimization on finite sets
Seminar - 11th Floor Lecture Hall
• Speaker
• Mauricio Velasco, Universidad de Los Andes
• Session Chair
• Jesús De Loera, University of California, Davis
###### Abstract
If $X\subseteq \mathbb{R}^n$ is a finite set then every function on $X$ can be written as the restriction of a polynomial in n-variables. As a result, polynomial optimization on finite sets is literally the same as general (nonlinear) optimization on such sets. Thinking of functions as polynomials, however, provides us with plenty of additional structures which can be leveraged for constructing better (or at least different) optimization algorithms. In these lectures, we will overview some of the key problems and results coming from this algebraic point of view. Specifically, we will discuss: How to prove that a polynomial function is nonnegative on a finite set X? What kind of algebraic certificates (proofs) are available and what can we say about their size and complexity? If the set $X$ has symmetries, can we leverage them in some systematic way that is useful for optimization? Characterizing the affine linear functions that are nonnegative on $X$ gives a description of the polytope $P={\rm Conv}(X)$. Stratifying such functions by the degree of their nonnegativity certificates leads to (semidefinite) hierarchies of approximation for the polytope $P$ and it is natural to ask about their speed of convergence and its relationship with the combinatorics of $P$ Finally, if time permits we will discuss some recent ideas combining the above methods with reinforcement learning as a way to improve scalability for combinatorial optimization problems. The results in (1),(2), (3) above are due to Blekherman, Gouveia, Laurent, Nie, Parrilo, Saunderson, Thomas, and others. These lectures intend to be a self-contained introduction to this vibrant and exciting research area.
• 11:00 - 11:30 am EST
Coffee Break
11th Floor Collaborative Space
• 11:30 am - 12:30 pm EST
Polynomial optimization on finite sets
Seminar - 11th Floor Lecture Hall
• Speaker
• Mauricio Velasco, Universidad de Los Andes
• Session Chair
• Jesús De Loera, University of California, Davis
###### Abstract
If $X\subseteq \mathbb{R}^n$ is a finite set then every function on $X$ can be written as the restriction of a polynomial in n-variables. As a result, polynomial optimization on finite sets is literally the same as general (nonlinear) optimization on such sets. Thinking of functions as polynomials, however, provides us with plenty of additional structures which can be leveraged for constructing better (or at least different) optimization algorithms. In these lectures, we will overview some of the key problems and results coming from this algebraic point of view. Specifically, we will discuss: How to prove that a polynomial function is nonnegative on a finite set X? What kind of algebraic certificates (proofs) are available and what can we say about their size and complexity? If the set $X$ has symmetries, can we leverage them in some systematic way that is useful for optimization? Characterizing the affine linear functions that are nonnegative on $X$ gives a description of the polytope $P={\rm Conv}(X)$. Stratifying such functions by the degree of their nonnegativity certificates leads to (semidefinite) hierarchies of approximation for the polytope $P$ and it is natural to ask about their speed of convergence and its relationship with the combinatorics of $P$ Finally, if time permits we will discuss some recent ideas combining the above methods with reinforcement learning as a way to improve scalability for combinatorial optimization problems. The results in (1),(2), (3) above are due to Blekherman, Gouveia, Laurent, Nie, Parrilo, Saunderson, Thomas, and others. These lectures intend to be a self-contained introduction to this vibrant and exciting research area.
• 12:30 - 2:30 pm EST
Lunch/Free Time
• 3:30 - 4:00 pm EST
Coffee Break
11th Floor Collaborative Space

All event times are listed in ICERM local time in Providence, RI (Eastern Standard Time / UTC-5).