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We investigate the computational complexity of a general "compression task" centrally occurring in the recently developed technique of iterative compression for exactly solving NP-hard minimization problems. The core issue (particularly but not only motivated by iterative compression) is to determine the computational complexity of, given an already inclusion-minimal solution for an underlying (typically NPhard) vertex deletion problem in graphs, to find a better disjoint solution. Thedoi:10.1145/1944857.1944860 fatcat:ci3bfpo4r5bulg3g4t4y2n2xly