Completeness, approximability and exponential time results for counting problems with easy decision version

Elementary Algorithmic Methods for Solving Suguru Puzzles

“…end if end if 32: end for 33: return ρ the Yin-Yang puzzle has a tractable variant when considering puzzles of size m × n with m ≤ 2 or n ≤ 2, where the number of solutions is correspondingly bounded above by O(n) or O(m) [21, Theorem 2 and Theorem 3], enabling the retrieval of all solutions in polynomial time. Nonetheless, in the subsequent analysis, we demonstrate that for any arbitrary 1 × n Suguru instance with H hints, the number of solutions is factorial in terms of n − H. It is not unexpected that this observation holds, considering that some computationally easy decision problems (with polynomial time complexities) show non-polynomial time complexities when their corresponding counting problems are considered [38].…”

Note on Algorithmic Investigations of Juosan Puzzles

“…We also mathematically investigated the number of solutions for Juosan puzzles of size 1 × n, which has an upper bound of O(( 13 (γ + + γ − + 1)) n ) where γ ± = 3 19 ± 3 √ 33 or roughly O(1.8392 n ) as shown in Theorem 4. This outcome is not unexpected given that some easy decision problems exhibit exponential time complexity for their corresponding counting problems [30]. Notably, our current upper bound for the number of solutions of a numbered territory within a 1 × n Juosan puzzle is not very tight, as finding a closed form or a tight upper bound for the found formula is challenging.…”

“…However, this algorithm does not recover all possible solutions to a Juosan instance of size 1 × n. In some puzzles, such as Yin-Yang puzzles of size 1×n and 2×n, the number of solutions to these puzzles is bounded by O(n) (see [23,Theorem 2 and Theorem 3]), and thus discovering all solutions to these instances can be done in polynomial time. Nevertheless, in the subsequent analysis, we shall show that the number of solutions to any arbitrary Juosan instance of size 1 × n is exponential in terms of n. This is unsurprising because some easy decision problems have exponential time results for their corresponding counting problems [30].…”

“…Investigation of tractable sub-problems and tractable variants of NP-complete problems are particularly important in theoretical computer science (see, e.g., [27,28]). Moreover, counting the number of solutions to a particular computational problem is also interesting from mathematical and computational perspectives, especially in counting complexity theory [29,30].…”