NP-Completeness of Fill-a-Pix and Σ&lt;i&gt;&lt;sup&gt;P&lt;/sup&gt;&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;-Completeness of Its Fewest Clues Problem

Higuchi, Yuta; Kimura, Kei

doi:10.1587/transfun.e102.a.1490

Cited by 1 publication

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“…Also FCP versions of several pencil-and-paper puzzles, including SUDOKU are Σ P 2 -complete. Since the setting of FCP is suitable for puzzles, FCP versions of puzzles are investigated (Higuchi and Kimura 2019;Goergen et al 2022).…”

Section: Related Workmentioning

confidence: 99%

Theoretical Aspects of Generating Instances with Unique Solutions: Pre-assignment Models for Unique Vertex Cover

Horiyama,

Kobayashi,

Ono

et al. 2024

AAAI

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The uniqueness of an optimal solution to a combinatorial optimization problem attracts many fields of researchers' attention because it has a wide range of applications, it is related to important classes in computational complexity, and the existence of only one solution is often critical for algorithm designs in theory. However, as the authors know, there is no major benchmark set consisting of only instances with unique solutions, and no algorithm generating instances with unique solutions is known; a systematic approach to getting a problem instance guaranteed having a unique solution would be helpful. A possible approach is as follows: Given a problem instance, we specify a small part of a solution in advance so that only one optimal solution meets the specification. This paper formulates such a ``pre-assignment'' approach for the vertex cover problem as a typical combinatorial optimization problem and discusses its computational complexity. First, we show that the problem is ΣP2-complete in general, while the problem becomes NP-complete when an input graph is bipartite. We then present an O(2.1996^n)-time algorithm for general graphs and an O(1.9181^n)-time algorithm for bipartite graphs, where n is the number of vertices. The latter is based on an FPT algorithm with O*(3.6791^τ) time for vertex cover number τ. Furthermore, we show that the problem for trees can be solved in O(1.4143^n) time.

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Section: Related Workmentioning

confidence: 99%