Abstract. The question to enumerate all (inclusionwise) minimal connected dominating sets in a graph of order [Formula: see text] in time significantly less than [Formula: see text] is an open question that was asked in many places. We answer this question affirmatively, by providing an enumeration algorithm that runs in time [Formula: see text], using polynomial space only. The key to this result is the consideration of this enumeration problem on 2-degenerate graphs, which is proven to be possible in time [Formula: see text]. Apart from solving this old open question, we also show new lower bound results. More precisely, we construct a family of graphs of order [Formula: see text] with [Formula: see text] many minimal connected dominating sets, while previous examples achieved [Formula: see text]. Our example happens to yield 4-degenerate graphs. Additionally, we give lower bounds for the previously not considered classes of 2-degenerate and of 3-degenerate graphs, which are [Formula: see text] and [Formula: see text], respectively. We also address essential questions concerning output-sensitive enumeration. Namely, we give reasons why our algorithm cannot be turned into an enumeration algorithm that guarantees polynomial delay without much effort. More precisely, we prove that it is NP -complete to decide, given a graph [Formula: see text] and a vertex set [Formula: see text], if there exists a minimal connected dominating set [Formula: see text] with [Formula: see text], even if [Formula: see text] is known to be 2-degenerate. Our reduction also shows that even any subexponential delay is not easy to achieve for enumerating minimal connected dominating sets. Another reduction shows that no FPT -algorithms can be expected for this extension problem concerning minimal connected dominating sets, parameterized by [Formula: see text]. This also adds one more problem to the still rather few natural parameterized problems that are complete for the parameterized complexity class W [3]. We also relate our enumeration problem to the famous Hitting Set Transversal problem, a problem open for more than four decades, which can be phrased in our context as the question to enumerate all minimal dominating sets of a graph with polynomial delay, by showing that a polynomial-delay enumeration algorithm for minimal connected dominating sets implies an affirmative algorithmic (polynomial-delay) solution to the Hitting Set Transversal problem.

The question to enumerate all (inclusion-wise) minimal connected dominating sets in a graph of order n in time significantly less than 2ⁿ is an open question that was asked in many places. We answer this question affirmatively, by providing an enumeration algorithm that runs in time 𝒪(1.9896ⁿ), using polynomial space only. The key to this result is the consideration of this enumeration problem on 2-degenerate graphs, which is proven to be possible in time 𝒪(1.9767ⁿ). Apart from solving this old open question, we also show new lower bound results. More precisely, we construct a family of graphs of order n with Ω(1.4890ⁿ) many minimal connected dominating sets, while previous examples achieved Ω(1.4422ⁿ). Our example happens to yield 4-degenerate graphs. Additionally, we give lower bounds for the previously not considered classes of 2-degenerate and of 3-degenerate graphs, which are Ω(1.3195ⁿ) and Ω(1.4723ⁿ), respectively. We also address essential questions concerning output-sensitive enumeration. Namely, we give reasons why our algorithm cannot be turned into an enumeration algorithm that guarantees polynomial delay without much efforts. More precisely, we prove that it is NP-complete to decide, given a graph G and a vertex set U, if there exists a minimal connected dominating set D with U ⊆ D, even if G is known to be 2-degenerate. Our reduction also shows that even any subexponential delay is not easy to achieve for enumerating minimal connected dominating sets. Another reduction shows that no FPT-algorithms can be expected for this extension problem concerning minimal connected dominating sets, parameterized by |U|. This also adds one more problem to the still rather few natural parameterized problems that are complete for the class W[3]. We also relate our enumeration problem to the famous open Hitting Set Transversal problem, which can be phrased in our context as the question to enumerate all minimal dominating sets of a graph with polynomial delay by showing that a polynomial-delay enumeration algorithm for minimal connected dominating sets implies an affirmative algorithmic solution to the Hitting Set Transversal problem.

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 NP-hard and the optimisation variant does not allow a polynomial-time approximation scheme, unless P = NP. 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 NP-complete (and its optimisation version is APX-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 $\mathcal {O}(3^{\mid {w}\mid })$ O ( 3 ∣ w ∣ ) 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.

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(3n) 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).