In one of his early works, Samuel Huntington, Professor Emeritus at Harvard, argued that corruption is a preferred way to achieve efficiency in society which by far surpasses a revolutionary overthrow of a current system (I did not read it myself but that's what Robert Kaplan in his article about SH implies - see Atlantic Monthly, December 2001).
Bribing a police officer to get things done can be in the long term and in certain cases better for the society than if people decide to take the police stations by force. No matter how far-fetched this argument might be, in corruption we're facing a phenomenon which carries different perceptions in Europe and in the US.
Politically, I don't think you can eradicate the problem without solving the issue of financing of political parties. As long as we have parties emerging and disappearing without an independent or secured source of income and at the speed of 2-3 parties per election cycle, one cannot expect that Slovakia or any other country will be free of corruption at the highest levels for some time to come.
Improved economic conditions can and will take care of the small stuff, but painting over systemic shortcomings in the political system can seriously undermine the public's trust in their representatives and thus open the door for populists of all kinds.
Unfortunately, I don't see any model for Slovakia other than to take the problem head on. If you have a better one, please pass it on.