NP-Completeness of Reallocation Problems with Restricted Block Volume

A reallocation problem is defined as determining whether blocks can be moved from their current boxes to their destination boxes in the number of moves less than or equal to a given positive integer. This problem is defined simply and has many practical applications. We previously proved the following results: The reallocation problem such that the block volume is restricted to one volume unit for all blocks can be solved in linear time. But the reallocation problem such that the block volume is not restricted is NP-complete, even if no blocks have bypass boxes. But the computational complexity of the reallocation problems such that the range of the block volume is restricted has not yet been known. We prove that the reallocation problem is NP-complete even if the block volume is restricted to one or two volume units for all blocks and no blocks have bypass boxes. Furthermore, we prove that the reallocation problem is NP-complete, even if there are only two blocks such that each block has the volume units of fixed positive integer greater or equal than two, the volume of the other blocks is restricted to one volume unit, and no blocks have bypass boxes. Next, we consider such a block that must stays in a same box both in the initial state and in the final state but can be moved to another box for making room for other blocks. We prove that the reallocation problem such that an instance has such blocks is also NP-complete even if the block volume is restricted to one volume unit for all blocks. key words: reallocation, computational complexity, NP-