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Complexity and Completeness of Finding Another Solution and Its Application to Puzzles
Takayuki YATO Takahiro SETA
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Publication Date: 2003/05/01
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Discrete Mathematics and Its Applications)
computational complexity, NP-complete, another solution, puzzle,
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The Another Solution Problem (ASP) of a problem is the following problem: for a given instance x of and a solution s to it, find a solution to x other than s. The notion of ASP as a new class of problems was first introduced by Ueda and Nagao. They also pointed out that parsimonious reductions which allow polynomial-time transformation of solutions can derive the NP-completeness of ASP of a certain problem from that of ASP of another. In this paper we consider n-ASP, the problem to find another solution when n solutions are given, and formalize it to investigate its characteristics. In particular we consider ASP-completeness, the completeness with respect to the reductions satisfying the properties mentioned above. The complexity of ASPs has a relation with the difficulty of designing puzzles. We prove the ASP-completeness of three popular puzzles: Slither Link, Cross Sum, and Number Place. Since ASP-completeness implies NP-completeness, these results can be regarded as new results of NP-completeness proof of puzzles.