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This document contains description of the talks at the Dagstuhl seminar 16451 "Structure and Hardness in P". The main goal of the seminar was to bring together researchers from several disciplines and connect those who work on proving conditional lower bounds with those who or may benefit from it. This resulted in an extensive list of open problems which is also provided.doi:10.4230/dagrep.6.11.1 dblp:journals/dagstuhl-reports/LewensteinPW16 fatcat:hha7ev23anezhi37biylsyfxie