I'm referring to a question in this book: Algorithms by Jeff Erickson, link:http://jeffe.cs.illinois.edu/teaching/algorithms/notes/J-approx.pdf, in particular, pg.21, Q11.

The author mentioned that finding the chromatic number $\chi(G)$ is np-hard, and ask to show that return any integer between $\chi(G)$ and $\chi(G) + 31337$ is also np-hard. I believe the constant doesn't matter here, my approach is as following: given an instance of finding the chromatic number, we can modify it to finding $\chi(G)$ and $\chi(G) + c$, return an integer between $\chi(G)$ and $\chi(G) + c$.

From one suggestion seems like I'm having a wrong approach, any hint that helps me go further is appreciated.

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    – D.W.
    Dec 6, 2021 at 5:31

1 Answer 1


No. Your reduction goes the wrong way. You will need to find a different proof.

It might be easy to find an algorithm that returns any integer between $\chi(G)$ and $\chi(X)+31337$, and hard to find an algorithm that returns exactly $\chi(G)$. (It isn't, but that isn't obvious, and you'll have to prove it.)

Analogy: it's easy for you to find an integer between my age and my age plus 31337, but I suspect it'd be hard for you to identify my age exactly. Try applying your proposed logic to this analogous situation, and hopefully you'll see what goes wrong.

  • $\begingroup$ So apparently what goes wrong is we knew finding an exact range is harder than finding a number in this range, but it turns out we need to prove finding a number in this range is as hard as finding the range? $\endgroup$
    – hh vh
    Dec 6, 2021 at 6:45
  • $\begingroup$ @hhvh, I'm unsure what you mean by "finding an exact range". $\chi(G)$ isn't a range -- it is a single number. I suggest spending some more time working through the details as outlined in my answer. $\endgroup$
    – D.W.
    Dec 6, 2021 at 9:03

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