On the Complexity of Grammar-Based Compression over Fixed Alphabets

It is shown that the shortest-grammar problem remains NP-complete if the alphabet is fixed and has a size of at least 24 (which settles an open question). On the other hand, this problem can be solved in polynomial-time, if the number of nonterminals is bounded, which is shown by encoding the problem as a problem on graphs with interval structure. Furthermore, we present an O(3) exact exponential-time algorithm, based on dynamic programming. Similar results are also given for 1-level grammars, i. e., grammars for which only the start rule contains nonterminals on the right side (thus, investigating the impact of the “hierarchical depth” on the complexity of the shortest-grammar problem). 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, E.4 Coding and Information Theory