Periodic Reporting for period 1 - CoSP (Combinatorial Structures and Processes)
Periodo di rendicontazione: 2019-01-01 al 2020-12-31
CoSP addresses international collaboration in discrete mathematics and theoretical computer science. Experts from different continents and different domains will work with each other and with students in various stages of their studies. This project aims to develop various skills increasing the career prospects of the involved researchers and students both in academia and in industry.
The fields of research are:
(a) Matching theory for graphs and hypergraphs,
(b) Algorithms and complexity,
(c) Graph homomorphisms.
We concentrate on graph theory - a basic field in combinatorics, that deals with connections between pairs of objects. This has innumerably many applications, from communication to kidney transplants (applications to the latter led to a recent Nobel prize) and to theoretical physics.
Specific lines of research:
• Understanding the mysteriously good behavior of the intersection of two matroids with respect to representation and coloring problems.
• Designing a (1+ε)-approximation algorithm for edit distance running in almost linear time.
• Proving a super-linear lower bound for circuits of logarithmic depth.
• Algorithmic approaches to coloring of random regular, large girth and Erdős–Rényi graphs.
• A long-standing conjecture called the Pentagon problem which states that all sub-cubic graphs of large girth are 5-circular colorable.
• Algorithmic approaches to the planted Travelling Salesman Problem with random weights on the edges.
• Algorithmic and combinatorial approaches to problems coming from statistical physics.
• The classification of classes of structures defined by forbidden homomorphisms in the context of Ramsey theory, model theory and topological dynamics.
A key part of our project is aimed at the exchange and training of the early stage researchers.
The fields of research are:
(a) Matching theory for graphs and hypergraphs,
(b) Algorithms and complexity,
(c) Graph homomorphisms.
We concentrate on graph theory - a basic field in combinatorics, that deals with connections between pairs of objects. This has innumerably many applications, from communication to kidney transplants (applications to the latter led to a recent Nobel prize) and to theoretical physics.
Specific lines of research:
• Understanding the mysteriously good behavior of the intersection of two matroids with respect to representation and coloring problems.
• Designing a (1+ε)-approximation algorithm for edit distance running in almost linear time.
• Proving a super-linear lower bound for circuits of logarithmic depth.
• Algorithmic approaches to coloring of random regular, large girth and Erdős–Rényi graphs.
• A long-standing conjecture called the Pentagon problem which states that all sub-cubic graphs of large girth are 5-circular colorable.
• Algorithmic approaches to the planted Travelling Salesman Problem with random weights on the edges.
• Algorithmic and combinatorial approaches to problems coming from statistical physics.
• The classification of classes of structures defined by forbidden homomorphisms in the context of Ramsey theory, model theory and topological dynamics.
A key part of our project is aimed at the exchange and training of the early stage researchers.
Work in the first work package was done mainly by secondments to Princeton University. During the secondments, we worked mainly with Maria Chudnovsky (Princeton University), Paul Seymour (Princeton University), Noga Alon (Princeton University) Sophie Spirkl, former doctoral student of Paul Seymour, and with Cemil Dibek, a current doctoral student of Maria Chudnovsky. Cemil later visited Charles University in 2019 on a CoSP secondment. In 2020, we discussed extensively the topological methods with the participants of the CoSP seminar on topological methods (remotely). We organized an online CoSP ZOOM seminar on topological combinatorics, June 30 - July 30 2020.
Work in the second package was done by secondments to Rutgers University and to Simon-Fraser University. We worked mainly with Michael Saks of Rutgers University and with Bojan Mohar, Matt DeVos and David Wood from Simon Fraser University. The collaboration on CoSP in CNRS took place remotely without secondments; however, several results were reached.
Work in the third work package was done by secondments at Simon Fraser University and at Rutgers University. We were working mostly with Bojan Mohar, Matt DeVos and Pavol Hell and with Gregory Cherlin of Rutgers University.
Work package 4 –Training. In two months of 2019 spent at RU-Dimacs our students worked on a research project of their choice, mentored by researchers from Rutgers and Princeton. As a side note, our project passes the Bechdel–Wallace test, one of the research groups was formed entirely by girls. The students worked on the following research areas: Compression, Biased random walks, A game problem in extremal combinatorics, Rainbow structures. We published a booklet of REU 2019.
In 2020, all training secondments had to be cancelled. The training was done successfully online without secondments implemented. The students worked on four projects: Antipodal monochromatic paths in hypercubes, High School Partitions, On the Optimal Starting State for a Deterministic Scan in the Three-Color Potts Model, ∆-coloring in the graph streaming model.
We organized in 2019 CoSP School on algorithms (Rutgers University, USA), CoSP School on topological methods (Charles University, CZ), CoSP School on homomorphisms (Charles University, CZ), CoSP student workshop (Charles University, CZ).
Year 2020 was stigmatized by COVID-19 pandemic. Besides online training of early stage researchers and CoSP ZOOM seminar on topological combinatorics, we organized online CoSP student workshop. Furthermore, we organized the CoSP midterm meeting, where researchers from Europe, America and the rest of the world participated in a series of online lectures.
Work in the second package was done by secondments to Rutgers University and to Simon-Fraser University. We worked mainly with Michael Saks of Rutgers University and with Bojan Mohar, Matt DeVos and David Wood from Simon Fraser University. The collaboration on CoSP in CNRS took place remotely without secondments; however, several results were reached.
Work in the third work package was done by secondments at Simon Fraser University and at Rutgers University. We were working mostly with Bojan Mohar, Matt DeVos and Pavol Hell and with Gregory Cherlin of Rutgers University.
Work package 4 –Training. In two months of 2019 spent at RU-Dimacs our students worked on a research project of their choice, mentored by researchers from Rutgers and Princeton. As a side note, our project passes the Bechdel–Wallace test, one of the research groups was formed entirely by girls. The students worked on the following research areas: Compression, Biased random walks, A game problem in extremal combinatorics, Rainbow structures. We published a booklet of REU 2019.
In 2020, all training secondments had to be cancelled. The training was done successfully online without secondments implemented. The students worked on four projects: Antipodal monochromatic paths in hypercubes, High School Partitions, On the Optimal Starting State for a Deterministic Scan in the Three-Color Potts Model, ∆-coloring in the graph streaming model.
We organized in 2019 CoSP School on algorithms (Rutgers University, USA), CoSP School on topological methods (Charles University, CZ), CoSP School on homomorphisms (Charles University, CZ), CoSP student workshop (Charles University, CZ).
Year 2020 was stigmatized by COVID-19 pandemic. Besides online training of early stage researchers and CoSP ZOOM seminar on topological combinatorics, we organized online CoSP student workshop. Furthermore, we organized the CoSP midterm meeting, where researchers from Europe, America and the rest of the world participated in a series of online lectures.
We mention only one result of each WP with possibly highest expected impact beyond the state of the art.
WP1. We made an important step towards an extensively studied conjecture of Durhuis. The motivation of this conjecture is from the theoretical physics, namely whether certain quantum field theory makes sense. The work of Chudnovsky, Loebl, Seymour established an important step towards a possible combinatorial resolution of the conjecture, namely:
Martin Loebl conjectured that there are only exponentially many different d-connected graphs with minimum degree d and bounded maximum degree (as a function of the number of vertices) avoiding a given d-connected topological minor H with maximum degree at most d. Martin Loebl with Paul Seymour and Maria Chudnovsky managed to completely solved this question in the paper M. Chudnovsky, M. Loebl, P. Seymour Small Families Under Subdivision (manuscript 2019).
WP2. Algorithmic properties of high dimensional inference problems are surrounded by many open questions. Lenka Zdeborová with collaborators managed to solve an important among those problems, leading to an article “Marvels and Pitfalls of the Langevin Algorithm in Noisy High-Dimensional Inference” that already appeared in a prestigious journal Physical Review X. Gradient-descent-based algorithms and their stochastic versions have widespread applications in machine learning and statistical inference. An analytic study of the performances of one of them is accomplished, namely the Langevin algorithm.
WP3. Jan Hubička and Matěj Konečný collaborated with Gregory Cherlin on the study of homogeneous structures, giving probably the strongest evidence for the completeness of Cherlin's catalogue of metrically homogeneous graphs.
Overall, we expect to fulfil the plan put forward in our proposal, in all the tasks.
WP1. We made an important step towards an extensively studied conjecture of Durhuis. The motivation of this conjecture is from the theoretical physics, namely whether certain quantum field theory makes sense. The work of Chudnovsky, Loebl, Seymour established an important step towards a possible combinatorial resolution of the conjecture, namely:
Martin Loebl conjectured that there are only exponentially many different d-connected graphs with minimum degree d and bounded maximum degree (as a function of the number of vertices) avoiding a given d-connected topological minor H with maximum degree at most d. Martin Loebl with Paul Seymour and Maria Chudnovsky managed to completely solved this question in the paper M. Chudnovsky, M. Loebl, P. Seymour Small Families Under Subdivision (manuscript 2019).
WP2. Algorithmic properties of high dimensional inference problems are surrounded by many open questions. Lenka Zdeborová with collaborators managed to solve an important among those problems, leading to an article “Marvels and Pitfalls of the Langevin Algorithm in Noisy High-Dimensional Inference” that already appeared in a prestigious journal Physical Review X. Gradient-descent-based algorithms and their stochastic versions have widespread applications in machine learning and statistical inference. An analytic study of the performances of one of them is accomplished, namely the Langevin algorithm.
WP3. Jan Hubička and Matěj Konečný collaborated with Gregory Cherlin on the study of homogeneous structures, giving probably the strongest evidence for the completeness of Cherlin's catalogue of metrically homogeneous graphs.
Overall, we expect to fulfil the plan put forward in our proposal, in all the tasks.