Algorithms for Automatic Ranking of Participants and Tasks in an Anonymized Contest

We introduce a new set of problems based on the Chain Editing problem. In our version of Chain Editing, we are given a set of anonymous participants and a set of undisclosed tasks that every participant attempts. For each participant-task pair, we know whether the participant has succeeded at the task or not. We assume that participants vary in their ability to solve tasks, and that tasks vary in their difficulty to be solved. In an ideal world, stronger participants should succeed at a superset of tasks that weaker participants succeed at. Similarly, easier tasks should be completed successfully by a superset of participants who succeed at harder tasks. In reality, it can happen that a stronger participant fails at a task that a weaker participants succeeds at. Our goal is to find a perfect nesting of the participant-task relations by flipping a minimum number of participant-task relations, implying such a "nearest perfect ordering" to be the one that is closest to the truth of participant strengths and task difficulties. Many variants of the problem are known to be NP-hard. We propose six natural $k$-near versions of the Chain Editing problem and classify their complexity. The input to a $k$-near Chain Editing problem includes an initial ordering of the participants (or tasks) that we are required to respect by moving each participant (or task) at most $k$ positions from the initial ordering. We obtain surprising results on the complexity of the six $k$-near problems: Five of the problems are polynomial-time solvable using dynamic programming, but one of them is NP-hard.