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Hopefully this question isn't too general, but I was wondering what the relationship is between randomized algorithms that perform well with high-probability and those that perform well in expectation. My question is motivated by the definition of a randomized $\alpha$-approximation algorithm given here, namely that it is a polynomial-time algorithm that produces a solution within $\alpha$ of OPT in expectation or with high probability. I also found that the first few pages of this source provides some good insight into the high-probability vs. expectation approaches, but I still have questions.

  • Can you always transform an algorithm that achieves an $\alpha$-approximation in expectation to one that achieves this with high probability, and vice versa? (Ostensibly by rerunning the algorithm multiple [a polynomial number] of times.)
  • If not, is one harder than the other to obtain? (I would think that if you fix $\alpha$, a high-probability algorithm would always be harder to find/less likely to exist. Or maybe you can always find one, but the approximation ratio will become worse.)

Thanks for the help!

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If you have an algorithm that is an $\alpha$-approximation in expectation, then you can construct an algorithm that is a $(1+\epsilon)\alpha$-approximation with high probability, for any $\epsilon>0$. In particular, by Markov's inequality, if you run the algorithm, then with probability at least $1-1/(1+\epsilon)$ it will output a $(1+\epsilon)\alpha$-approximation. So, if you run the algorithm about $(c \log n)/\epsilon$ times and keep the best output among all of those trials, with probability about $1-1/n^c$ you will find a $(1+\epsilon)\alpha$-approximation.

If you have an algorithm that is an $\alpha$-approximation with high probability, there are no guarantees about the expectation. It's possible that with very small probability (probability $1/n^c$), it outputs an extremely bad solution (one with exponentially large approximation factor), and in all other cases it outputs an $\alpha$-approximation. In this case, the expected value of the approximation factor will be very large, even though it has a very small probability to output such a bad solution.

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  • $\begingroup$ Thank you for this fantastic answer! I'm probably missing something obvious, but I am stuck at checking the $(c \log n) / \epsilon$. The probability of failure in one trial is at most $\frac{1}{1 + \epsilon}$, so we want to show that $$\left( \frac{1}{1 + \epsilon} \right)^{(c \log n) / \epsilon}$$ is small, i.e., about $1/n^c$. Clearly we are allowed to upper bound the expression. At first I was thinking you could apply $1 + \epsilon \le e^\epsilon$ in the denominator, but this would make the entire expression smaller, so it is a no go. What am I missing? $\endgroup$ – kanso37 Jul 13 '20 at 5:03
  • $\begingroup$ Also, does this result generalize to maximization problems as well? The problem being that Markov's inequality is now upper bounding the probability of the good event instead of the bad event. I tried to find some sort of "reverse Markov's inequality," but it seems that such inequalities are really ugly and not obviously applicable. $\endgroup$ – kanso37 Jul 13 '20 at 5:14
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    $\begingroup$ @kanso37, When $\epsilon$ is small, $1/(1+\epsilon)\approx 1-\epsilon$, and $(1-\epsilon)^{1/\epsilon} \approx 1/e$. You should be able to take it from there. I haven't thought about maximization algorithms. The answer might be different, if solutions have non-negative value. $\endgroup$ – D.W.♦ Jul 13 '20 at 6:02
  • $\begingroup$ Thank you so much! $\endgroup$ – kanso37 Jul 14 '20 at 1:39

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