Periodic Reporting for period 2 - SUBMODULAR (The Power of Randomness and Continuity in Submodular Optimization) Reporting period: 2021-05-01 to 2022-10-31 Summary of the context and overall objectives of the project This research project deals with the theoretical foundations of submodular maximization.Submodular maximization affects many aspects of our daily lives, and has found applications ranging from pollution detection, the spread of influence in social networks, and even reduction of gang violence.Thus, it is no surprise that optimization problems with a submodular objective have been the focus of intense theoretical and practical research for more than a decade.The overarching goal of this research is to enrich the algorithmic toolkit and devise new algorithmic approaches that can be broadly applied to fundamental problems in submodular maximization and combinatorial optimization. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The study of new methods for designing and analyzing combinatorial algorithms for submodular maximization problems has been initiated.One such example includes the maximization of a submodular function given a knapsack constraint (a classic optimization problem which has found numerous practical applications throughout the years).Moreover, problems relating to submodularity and other ascepts, i.e. fault tolerance and clustering, have also been the focus of research. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) We plan to study how randomness and continuous approaches can be used for submodular maximization and more broadly applied to fundamental problems in combinatorial optimization.The ultimate goal is to present such new algorithmic approaches that will enable us to resolve many of the basic submodular optimization problems.
