The fewest clues problem

Demaine, Erik D.; Ma, Fermi; Schvartzman, Ariel; Waingarten, Erik; Aaronson, Scott

doi:10.1016/j.tcs.2018.01.020

Cited by 4 publications

(4 citation statements)

References 9 publications

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“…In pencil puzzles such as SU-DOKU, an instance is supposed to have a unique solution as the answer. Inspired by them, Demaine et al (Demaine et al 2018) define FCP (Fewest Clues Problem) type problems including FCP-SAT and FCP-SUDOKU. The FCP-SAT is defined as follows: Let ϕ be a CNF formula with a set of boolean variables X.…”

Section: Related Workmentioning

confidence: 99%

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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%

“…Theorem 3 is shown by performing a polynomialreduction from FCP 1-IN-3 SAT, which is known to be Σ P 2complete (Demaine et al 2018).…”

Section: Complexitymentioning

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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“…Thus, the number of additional variables to be assigned k is preserved. As FCP Planar 3SAT is Σ P 2 -complete (see [16]), FCP Roma is Σ P 2 -hard. Conversely, in Section 3.1 of [16] it is shown that the fewest clues problem of every problem from NP is in Σ P 2 .…”

Section: Further Complexity-theoretic Consequencesmentioning

confidence: 99%

“…As FCP Planar 3SAT is Σ P 2 -complete (see [16]), FCP Roma is Σ P 2 -hard. Conversely, in Section 3.1 of [16] it is shown that the fewest clues problem of every problem from NP is in Σ P 2 . After exhibiting these complexity-theoretic limitations of our problem, it is interesting to see algorithms that can possibly meet the limitations.…”

Section: Further Complexity-theoretic Consequencesmentioning

confidence: 99%

All Paths Lead to Rome

Goergen¹,

Fernau²,

Oest³

et al. 2022

Preprint

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All roads lead to Rome is the core idea of the puzzle game Roma. It is played on an n × n grid consisting of quadratic cells. Those cells are grouped into boxes of at most four neighboring cells and are either filled, or to be filled, with arrows pointing in cardinal directions. The goal of the game is to fill the empty cells with arrows such that each box contains at most one arrow of each direction and regardless where we start, if we follow the arrows in the cells, we will always end up in the special Roma-cell. In this work, we study the computational complexity of the puzzle game Roma and show that completing a Roma board according to the rules is an NP-complete task, counting the number of valid completions is #Pcomplete, and determining the number of preset arrows needed to make the instance uniquely solvable is Σ P 2 -complete. We further show that the problem of completing a given Roma instance on an n × n board cannot be solved in time O 2 o(n) under ETH and give a matching dynamic programming algorithm based on the idea of Catalan structures. * Supported by Deutsche Forschungsgemeinschaft (DFG), project FE560/9-1.

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