Algorithms that handle change over time and space

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My goal over the next five years is to understand how algorithms can be designed to accommodate change, where change can take place before, during, or after a problem is solved. Depending on when the change occurs, it might result in modifications to the original input or in multiple solutions to a single input. The type of change can be either inherent in or external to the problem and the inputs. The areas of parameterized complexity and reconfiguration lend themselves well to investigations of approaches to solving problems in such settings.***As a real-life example, repairs to power stations might require change between two good placements of customers, placements in which the output of each station is sufficient to meet the needs of all customers it serves. To minimize disruption of service, customers are moved one by one, each change resulting in a good assignment. In general, reconfiguration models changes that occur after a problem has been solved; solutions to a specific input (here, good assignments) can be viewed as vertices in a reconfiguration graph, where an edge indicates that one solution can be transformed into another in a single step, for some definition of a step. Recent research results have considered various properties of the reconfiguration graph, such as connectivity and diameter, and include the problem of determining whether there is a path from solution S to solution T (known as S-T connectivity). ***A natural setting for the study of change is the framework of parameterized complexity. Here, one identifies parameters of a problem and attempts to develop an algorithm running in time polynomial in the size of the input but subject to a potentially larger function of the parameters. This approach can yield practical algorithms for otherwise intractable problems in situations when the parameters are small, or demonstrate that such algorithms are unlikely to exist. Parameters can measure how much inputs can differ, how much change can be allowed, or how much solutions can vary for cases in which change occurs before, during, or after a problem is solved, respectively. My recent results combine reconfiguration and parameterized complexity; here parameters under consideration include the length of the transformation from one solution into another, bounds on the sizes of solutions, and properties of the inputs.***My two-fold approach to the study of change encompasses both general properties of classes of problems and results optimized for specific settings chosen from a wide spectrum of diverse application areas, such as fundamental problems with applications to biology, voting protocols, and communication and social networks. I plan to form a unified framework of models that captures change in both inputs and outputs. Examination of the parameterized complexity of such models may yield practical solutions to real-life problems as well as a general understanding about the nature of change.