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The background is from a highly cited paper "Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming".

I know how to prove $\frac{2\theta}{\pi}\ge \rho(1-\cos \theta)$ for $\theta\in [0, \pi]$, where $\rho=0.87856$.

But I get stuck in the proof of a related inequality $2-\frac{2\theta}{\pi}\ge \rho(1+\cos \theta)$ for $\theta\in [0, \pi]$?

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From where does the constant $\rho$ come? – Beni Bogosel Jun 29 '11 at 20:18
Look at the page 1122 of the paper – Sunni Jun 29 '11 at 20:33

2 Answers 2

up vote 3 down vote accepted

If you take $$\frac{ 2 \theta }{ \pi } \geq \rho (1-\cos\theta)$$ as given, and observe that $$ \cos(\theta) = -\cos(\pi - \theta) $$ for $\theta \in [0,\pi]$, you just substitute $\pi-\theta$ for $\theta$ in the original equation to get: \begin{align} \frac{ 2 (\pi-\theta) }{ \pi } &\geq \rho (1-\cos(\pi-\theta)) \\ 2 - \frac{ 2 \theta }{ \pi } &\geq \rho (1+\cos\theta) \end{align}

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The derivative of a quotient makes it much more complicated.... – Sunni Jun 29 '11 at 20:29
Oh, there's a much easier way to get the second inequality from the first. – trutheality Jun 29 '11 at 20:50
That is just what I am looking for. – Sunni Jun 29 '11 at 21:02

So you can do a true brute force attack on it. Create the function $g(\theta ) := 2 - \frac{2 \theta }{\pi} - \rho (1 + \cos \theta)$. It's continuous. Take the derivative and find the minimum on $[0, \pi ]$. You'll find $g \geq 0$, which is what you wanted.

So I just did it on W|A. But are you looking for a better way to do it? Something more witty?

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+1. Yes, I am looking for a better way to do it. – Sunni Jun 29 '11 at 20:21
@Sunni: what does your proof of the first inequality look like? – mixedmath Jun 29 '11 at 20:22
Look at the page 1122 of the paper – Sunni Jun 29 '11 at 20:26

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