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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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  • $\begingroup$ We discourage "please check whether my answer is correct" questions, as only "yes/no" answers are possible, which won't help you or future visitors. See here and here. Can you edit your post to ask about a specific conceptual issue you're uncertain about? As a rule of thumb, a good conceptual question should be useful even to someone who isn't looking at the problem you happen to be working on. If you just need someone to check your work, you might seek out a friend, classmate, or teacher. $\endgroup$
    – D.W.
    Dec 6, 2021 at 5:31

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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.

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  • $\begingroup$ 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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