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We say that a language $X$ is polynomial time reducible to $Y$, intuitively, if given an algorithm for solving $Y$, there's an algorithm for solving $X$. I know this can be formalized using Karp reductions or Cook reductions. I was wondering if there's a reduction that also acts on witnesses, in the following sense:

$f,g$ should be polynomial-time computable functions such that $(x,w)$ is in the NP-Relation for $X$ if and only if $(f(y),g(w))$ is in the NP-Relation for $Y$.

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    $\begingroup$ I don’t know what your goal is with these reductions. But there is a standard notion of many-one reductions for NP-search problems where $g$ goes in the opposite direction: a reduction of the search problem associated with $R(x,y)$ to the search problem associated with $S(w,z)$ is a pair of poly-time functions $f$ and $g$ such that $S(f(x),z)\implies R(x,g(x,z))$. This formalizes the idea that if we can efficiently find witnesses for $S$, we can efficiently find fitnesses for $R$ by first applying $f$, then searching for a witness for $S$, and then applying $g$ to get back a witness for $R$. $\endgroup$ Mar 21 at 10:03
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    $\begingroup$ (i) Parsimonious reductions are similar in spirit to what you have in mind. These are reductions that preserve the number of witnesses. E.g., a reduction from SAT to 3-COLOR so that the number of 3-colorings in the produced instance equals the number of satisfying assignments to the given instance. $\endgroup$
    – Neal Young
    Mar 22 at 0:09
  • $\begingroup$ (ii) I think the (standard) polytime reductions that I know of generally have the property you are asking for. This is usually implicit in the proof.. , which proves that "if $x$ is in $X$ then $f(x)$ is in $Y$", which is the same as "if $x$ has a witness $w$ then $f(x)$ has a witness $g(x, w)$". usually the function $g(x, w)$ is computable in polynomial time. strictly speaking, $g$ is not usually a function of $(x, w)$, not just $w$, but this is sort of a technicality, as $w$ could easily encode $x$. $\endgroup$
    – Neal Young
    Mar 22 at 0:12
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    $\begingroup$ @EmilJeřábek you need add that $f$ maps "yes" instances to "yes" instances, right? If the OP wants a reference, these reductions are mentioned in Goldreich's "Computational Complexity: a Conceptual Perspective", where he calls them Levin reductions. $\endgroup$ Mar 22 at 7:57
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    $\begingroup$ @DamianoMazza I guess you are right. I’m used to these reductions in the context of TFNP problems, in which case there are no “no” instances. $\endgroup$ Mar 22 at 12:41

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