We have obtained fast algorithms for subgraph-finding problems, in which the nodes of the distributed network need to detect instances of a certain type of subgraph. Prime examples are triangles or larger cliques, as well as small cycles. As a main methodology for this, we showed how to multiply sparse matrices fast, as well as compute other tasks over sparse inputs, which we eventually leveraged towards subgraph finding. We further showed how to obtain fast subgraph finding in a deterministic manner, by developing new tools for such distributed settings. In addition, we characterized the bandwidth needed for such tasks in a dynamic setting, in which the input graph continuously changes over time. In a yet harsher dynamic setting which assumes no restrictions at all on the topology changes, we show how to maintain information about all cliques in the network. Many of our results are accompanied by lower bounds, establishing the optimality of many of our algorithms.
We designed algorithms for symmetry-breaking problems, a prime example being reconfiguration of maximal independent sets (MIS). Moreover, we have designed fast algorithms for many variants of the classic coloring problem, and we obtained fast derandomization results, which resulted in fast deterministic MIS algorithms. We developed constant-time algorithms for maintaining solutions for symmetry-breaking tasks in the aforementioned highly-dynamic setting.
We addressed local optimization, and towards this goal we designed fast algorithms for approximating minimum vertex cover and minimum dominating sets on power graphs. Our research has also yielded fast algorithms for global optimization problems such as spanners and connectivity augmentation problems under restricted bandwidth. Furthermore, we studied hardness of local approximation and obtained various lower bounds, which also include novel techniques for these restricted bandwidth settings.
We have managed to make important connections to additional restricted bandwidth settings, showing connections between various settings in which the common thread is that the communication bandwidth is limited. Examples include several highly dynamic settings, a recently introduced hybrid model that combines global and local communication, a distributed quantum setting which models the exchange of qubits rather than bits along communication channels, and more.
