Periodic Reporting for period 1 - GRAPHCOSY (GRAPH reconstruction, COspectrality and SYnchronisation through the lens of number theory, geometry and algorithms)
Berichtszeitraum: 2022-07-01 bis 2024-06-30
The project studied various questions related to central problems in combinatorics, with the goal of gaining further insight into them. All of the questions could be motivated from algorithmic applications, but it is a project in pure mathematics of theoretical nature.
(1) Graph reconstruction. Reconstruction problems are everywhere in life: recovering a broken vase from its pieces, reconstructing the map of an ancient city from archeological findings, or reconstructing evolutionary histories from bacterial taxonomy data. It is very natural to ask when a graph (or graph property) can be reconstructed from its `pieces'. The main objective of the first work package is as follows.
(O1) Develop new techniques to reconstruct global graph parameters from local information.
Towards this, several explicit open questions were proposed, the first relating to finding the smallest size induced subgraphs for which you can reconstruct a tree. I am still involved in two ongoing collaborations related to this.
There are two projects that I did finish relating to this objective: one on reconstructing a graph from distance queries and the other on reconstructing a graph from the collection of k-tuples which form a connected graph. The latter has already been accepted by WG, an international conference in graph theory.
(2) Graph isomorphism: when are two networks, which may be described in very different ways, actually the same ('isomorphic')? The goal of this subpart of the project was to study a deeper mathematical theory that holds for much more general objects called manifolds, and see if the situation can be severely simplified in the special case of graphs.
(O2) Prove the analogous result of [G. Cornelissen and N. Peyerimhoff. Twisted isospectrality, homological wideness and isometry] for graphs: for a pair of cospectral graphs, give a simple algorithmic construction of finitely many matrices whose spectra determine whether or not the graphs are isomorphic.
I had several meetings with the supervisor and his co-author and indeed the situation seems to simplify quite a lot for graphs. This project is not yet finished and is still in the 'research' phase.
(3) Cerny's conjecture: what is the smallest length of a reset word, an instruction with the property that no matter where you start in a network, you always end up in the same location?
(O3) Improve our understanding of the smallest reset word of synchronising automata.
This has one part relating to what happens for a random automaton (3b) and one related to parameterized complexity (3a). For the random case, during the first month my project started, the upper bound got settled by an independent group of researchers (https://arxiv.org/pdf/2207.14108.pdf) with whom I then discussed the lower bound a bit. Before their work got put online, together with the two junior researchers I planned to work on this with during the proposal, we had obtained a weaker result but with simpler methods (I started working on this already before the start of the grant). I would have still written this down as a blogpost for this grant, since our methods were different than theirs, but before this could happen there was another researcher who 'scooped' us on this also.
We did gain more insight into the parameterized complexity of this problem and various relating problems, together with the second supervisor and a visitor.
(1) Graph reconstruction. Reconstruction problems are everywhere in life: recovering a broken vase from its pieces, reconstructing the map of an ancient city from archeological findings, or reconstructing evolutionary histories from bacterial taxonomy data. It is very natural to ask when a graph (or graph property) can be reconstructed from its `pieces'. The main objective of the first work package is as follows.
(O1) Develop new techniques to reconstruct global graph parameters from local information.
Towards this, several explicit open questions were proposed, the first relating to finding the smallest size induced subgraphs for which you can reconstruct a tree. I am still involved in two ongoing collaborations related to this.
There are two projects that I did finish relating to this objective: one on reconstructing a graph from distance queries and the other on reconstructing a graph from the collection of k-tuples which form a connected graph. The latter has already been accepted by WG, an international conference in graph theory.
(2) Graph isomorphism: when are two networks, which may be described in very different ways, actually the same ('isomorphic')? The goal of this subpart of the project was to study a deeper mathematical theory that holds for much more general objects called manifolds, and see if the situation can be severely simplified in the special case of graphs.
(O2) Prove the analogous result of [G. Cornelissen and N. Peyerimhoff. Twisted isospectrality, homological wideness and isometry] for graphs: for a pair of cospectral graphs, give a simple algorithmic construction of finitely many matrices whose spectra determine whether or not the graphs are isomorphic.
I had several meetings with the supervisor and his co-author and indeed the situation seems to simplify quite a lot for graphs. This project is not yet finished and is still in the 'research' phase.
(3) Cerny's conjecture: what is the smallest length of a reset word, an instruction with the property that no matter where you start in a network, you always end up in the same location?
(O3) Improve our understanding of the smallest reset word of synchronising automata.
This has one part relating to what happens for a random automaton (3b) and one related to parameterized complexity (3a). For the random case, during the first month my project started, the upper bound got settled by an independent group of researchers (https://arxiv.org/pdf/2207.14108.pdf) with whom I then discussed the lower bound a bit. Before their work got put online, together with the two junior researchers I planned to work on this with during the proposal, we had obtained a weaker result but with simpler methods (I started working on this already before the start of the grant). I would have still written this down as a blogpost for this grant, since our methods were different than theirs, but before this could happen there was another researcher who 'scooped' us on this also.
We did gain more insight into the parameterized complexity of this problem and various relating problems, together with the second supervisor and a visitor.
I worked on a new reconstruction model introduced by a group of researchers and proved various new mathematical and algorithmic results. This has already resulted in a conference paper in WG. During research visits, I worked on various questions relating to graph reconstruction that are still work-in-progress. We managed to answer a question which had come up naturally when trying to reconstruct degree sequences, namely, how many different degree sequences are there to begin with. This is probably one of my strongest results that I obtained: we send this to one of the top mathematical journals, where it is currently under review.
Relating to (O3a), we solved one of the questions I posed in my initial grant, but two other groups of researchers independently obtained similar results, and we therefore decided to not pursue ours.
Relating to (O3b), we found out the parameterized complexity will be in between various classes. While doing the inverstigations, we were able to pinpoint the complexity for various other problems, leading to a publication at ICALP, the flagship European conference for algorithms.
We also had several preliminary discussions relating to (O2).
I also find it important to transfer our knowledge as mathematicians to different fields. Via a previous supervisor, I helped a PhD student with the mathematical background on one of his projects, and he then asked me to be a co-author on the following paper which has already been accepted by a top Artificial Intelligence conference (AAMAS).
Relating to (O3a), we solved one of the questions I posed in my initial grant, but two other groups of researchers independently obtained similar results, and we therefore decided to not pursue ours.
Relating to (O3b), we found out the parameterized complexity will be in between various classes. While doing the inverstigations, we were able to pinpoint the complexity for various other problems, leading to a publication at ICALP, the flagship European conference for algorithms.
We also had several preliminary discussions relating to (O2).
I also find it important to transfer our knowledge as mathematicians to different fields. Via a previous supervisor, I helped a PhD student with the mathematical background on one of his projects, and he then asked me to be a co-author on the following paper which has already been accepted by a top Artificial Intelligence conference (AAMAS).
This is a project in theoretical mathematics. This means that all of my works proved new results which beat the state-of-the-art, and that in the long-term, my findings may have a big impact, but that there are no direct societal consequences.
Particularly for my project, besides expanding our knowledge I also transfer ideas between mathematics and computer science. The involved researchers will continue to disseminate this knowledge themselves.
Particularly for my project, besides expanding our knowledge I also transfer ideas between mathematics and computer science. The involved researchers will continue to disseminate this knowledge themselves.