Periodic Reporting for period 2 - CoSP (Combinatorial Structures and Processes)
Berichtszeitraum: 2021-01-01 bis 2024-07-31
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 and Cemil Dibek, a current doctoral student of Maria Chudnovsky. Ron Aharoni in collaboration with other authors proved lower bounds on the matching number of certain hypergraphs by using a topological version of Hall's theorem. These bounds yield rainbow versions of the KKM theorem for products of simplices, which in turn are used to obtain some results on multiple-cake division, and on rainbow matchings in families of d-intervals. Furthermore, CoSP has interesting results on rainbow structures. Furthermore, Ron Holzman improved the previously known results of generalizations of Erdős–Spencer theorem.
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. We addressed the cryptographic problem of designing oblivious RAM with partial progress in understanding the problems. Another cutting edge research studied a close connection between circuit complexity of sorting and a well-known conjecture on network coding. We explored a new research direction: algorithms, optimization and dynamics in the Algorithmic Game Theory. This is an exciting attractive and robust area to continue working on.
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. Within this research package, we worked on a counting problem for group connectivity. Furthermore, Jan Bok in collaboration with prof. Hell, Jedličková and other collaborators obtained a partial dichotomy for the list homomorphism problem for signed graphs.
Work package 4 –Training. Each year, CoSP organized a summer research experience and training for young talented researchers. Sixteeen to twenty participants per year conducted research on current problems in collaboration with top-grade researchers from the USA. For some participants, CoSP provided the first opportunity to publish a research paper and motivated them to pursue academic career. The training included also technical lectures and soft-skills training (scientific writing, ethics, research skills…). CoSP trained in total 76 talented young researchers between 2019 and 2024.
Events organized by CoSP: CoSP School on algorithms (Rutgers University, USA, 2019), CoSP School on topological methods (Charles University, CZ, 2019), CoSP School on homomorphisms (Charles University, CZ, 2019), CoSP school+workshop on matchings (Charles University, CZ, 2022), CoSP workshop on algorithms (Rutgers University, USA, 2024), CoSP ZOOM seminar on topological combinatorics, yearly CoSP student workshop (Charles University, CZ), CoSP annual, midterm and final meetings.
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. We addressed the cryptographic problem of designing oblivious RAM with partial progress in understanding the problems. Another cutting edge research studied a close connection between circuit complexity of sorting and a well-known conjecture on network coding. We explored a new research direction: algorithms, optimization and dynamics in the Algorithmic Game Theory. This is an exciting attractive and robust area to continue working on.
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. Within this research package, we worked on a counting problem for group connectivity. Furthermore, Jan Bok in collaboration with prof. Hell, Jedličková and other collaborators obtained a partial dichotomy for the list homomorphism problem for signed graphs.
Work package 4 –Training. Each year, CoSP organized a summer research experience and training for young talented researchers. Sixteeen to twenty participants per year conducted research on current problems in collaboration with top-grade researchers from the USA. For some participants, CoSP provided the first opportunity to publish a research paper and motivated them to pursue academic career. The training included also technical lectures and soft-skills training (scientific writing, ethics, research skills…). CoSP trained in total 76 talented young researchers between 2019 and 2024.
Events organized by CoSP: CoSP School on algorithms (Rutgers University, USA, 2019), CoSP School on topological methods (Charles University, CZ, 2019), CoSP School on homomorphisms (Charles University, CZ, 2019), CoSP school+workshop on matchings (Charles University, CZ, 2022), CoSP workshop on algorithms (Rutgers University, USA, 2024), CoSP ZOOM seminar on topological combinatorics, yearly CoSP student workshop (Charles University, CZ), CoSP annual, midterm and final meetings.
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). Currently, we explore the possibility that the graphs crucial for the Durhuis conjecture do not contain K4,4 as a topological minor. If true, using the previous result, the whole conjecrure of Durhuis follows.
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 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.
Research outcomes, which have been generated until now, are mainly for scientific use. However, as written above, the research of CoSP evolved into Algorithmic Game Theory, where the acquired methods and collaborations formed during CoSP find industrial and societal applications, e.g. in internet trading and societal logistics.
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). Currently, we explore the possibility that the graphs crucial for the Durhuis conjecture do not contain K4,4 as a topological minor. If true, using the previous result, the whole conjecrure of Durhuis follows.
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 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.
Research outcomes, which have been generated until now, are mainly for scientific use. However, as written above, the research of CoSP evolved into Algorithmic Game Theory, where the acquired methods and collaborations formed during CoSP find industrial and societal applications, e.g. in internet trading and societal logistics.