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    Polynomial-time reducibility preserves decision-problem s... — Carmelics
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    Challenges→The existence of a polynomial time algorithm for any single NP-complete problem would entail the existence of polynomial time algorithms for all problems in NP.

    Polynomial-time reducibility preserves decision-problem solvability in principle, but the reduction itself may introduce constant or hidden complexity factors that render the composed algorithm impractical even if formally polynomial.

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    1 reason for
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    Reasons For

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    Reason for
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    • 1.A 3SAT instance reduced to clique via polynomial reduction may require checking exponentially many cliques despite polynomial-bounded reduction overhead.
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    • 2.Constants hidden in Big-O notation (e.g., O(n^100)) can make formally polynomial algorithms computationally infeasible for all practical input sizes.
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    • 3.Chained reductions compound overhead multiplicatively: if each of k reductions adds factor c, total complexity multiplies by c^k even in polynomial class.
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    Reasons Against

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    • 1.Polynomial-time reducibility is defined precisely to avoid hidden complexity: if A reduces to B in poly-time, solving B solves A without exponential blowup.
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    • 2.Impracticality due to large constants is a separate issue from solvability-in-principle; conflating them misuses the theoretical concept of polynomial reduction.
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    • 3.Most natural NP-complete reductions (e.g., Cook-Levin) do introduce overhead, but this reflects problem hardness, not a flaw in reducibility as a formal tool.
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    Key Terms

    Composed algorithm(as used in computer science)
    A solution made by combining two or more methods together, where the output of one method becomes the input for the next.
    Decision problem(as used in computational complexity theory)
    A question you can only answer with 'yes' or 'no'—like 'does this number have a factor smaller than 5?'
    Hidden complexity factors(as used in computer science)
    Extra difficulty or slowness that isn't obvious at first glance, like needing to do a million tiny steps that individually seem fast but add up to a long time.
    Polynomial(as used in mathematics and computer science)
    A mathematical expression (like x² + 3x + 2) whose growth rate is reasonable and manageable, as opposed to exponential growth which explodes very quickly.
    Polynomial-time reducibility(as used in computer science and computational theory)
    A way of transforming one difficult problem into another difficult problem such that if you can solve the second one quickly, you can solve the first one quickly too.
    Solvability in principle(as used in logic and computation)
    The theoretical possibility that a problem *could* be solved, even if it might be impractical or take forever in reality.
    reduction(Lambda calculus or term-rewriting systems)
    A computational process analogous to computing the value of a function, proceeding through a series of discrete calculation steps applied to a term

    Connections

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    Truth & Knowledge1 linkedModality & Possibility1 linked

    Related

    A 3SAT instance reduced to clique via polynomial reduction may require checking ...Chained reductions compound overhead multiplicatively: if each of k reductions a...

    Details

    Type
    claim
    Perspectives
    2 (1 for, 1 against)
    Edits
    1 edit
    Constants hidden in Big-O notation (e.g., O(n^100)) can make formally polynomial...
    Impracticality due to large constants is a separate issue from solvability-in-pr...
    +3 moreShow less
    Most natural NP-complete reductions (e.g., Cook-Levin) do introduce overhead, bu...Polynomial-time reducibility is defined precisely to avoid hidden complexity: if...The existence of a polynomial time algorithm for any single NP-complete problem ...