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• W3098381330 abstract "Abstract In the smallest grammar problem, we are given a word w and we want to compute a preferably small context-free grammar G for the singleton language { w } (where the size of a grammar is the sum of the sizes of its rules, and the size of a rule is measured by the length of its right side). It is known that, for unbounded alphabets, the decision variant of this problem is N P -hard and the optimisation variant does not allow a polynomial-time approximation scheme, unless P = N P . We settle the long-standing open problem whether these hardness results also hold for the more realistic case of a constant-size alphabet. More precisely, it is shown that the smallest grammar problem remains N P -complete (and its optimisation version is A P X -hard), even if the alphabet is fixed and has size of at least 17. The corresponding reduction is robust in the sense that it also works for an alternative size-measure of grammars that is commonly used in the literature (i. e., a size measure also taking the number of rules into account), and it also allows to conclude that even computing the number of rules required by a smallest grammar is a hard problem. On the other hand, if the number of nonterminals (or, equivalently, the number of rules) is bounded by a constant, then the smallest grammar problem can be solved in polynomial time, which is shown by encoding it as a problem on graphs with interval structure. However, treating the number of rules as a parameter (in terms of parameterised complexity) yields W [1]-hardness. Furthermore, we present an $mathcal {O}(3^{mid {w}mid })$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>∣</mml:mo> <mml:mi>w</mml:mi> <mml:mo>∣</mml:mo> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:math> exact exponential-time algorithm, based on dynamic programming. These three main questions are also investigated 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” of grammars on the complexity of the smallest grammar problem. In this regard, we obtain for 1-level grammars similar, but slightly stronger results." @default.
• W3098381330 created "2020-11-23" @default.
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• W3098381330 date "2020-11-13" @default.
• W3098381330 modified "2023-10-12" @default.
• W3098381330 title "On the Complexity of the Smallest Grammar Problem over Fixed Alphabets" @default.
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• W3098381330 doi "https://doi.org/10.1007/s00224-020-10013-w" @default.
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