Doesn't relativity "solve" this this problem if we can reach near lightspeed, and don't mind it being a one way trip?
No, there is still an lower limit of 4.37 years to get to alpha-centauri within any frame of reference no matter how much you accelerate within conventional space.
What relativity does, is that if you travel close to speed of light time taken for the trip as measured on ship versus on earth could be different by many orders of magnitude.
edit: Was wrong, as per [0] "This is something many readers don't understand well, so it bears repeating: The journey times as experienced by those on the ship are not limited by the speed of light. Instead what they experience is the planetary reference frame getting relativistic."
[0] http://en.wikipedia.org/wiki/Space_travel_using_constant_acc...
For example, if you have a ship capable of continuous acceleration at 1g, you can reach any point in the universe on the order of ~50 years.
it would take a year to get to the speed of light at 1g (if my equation was right.) Then you can't speed up anymore.
It isn't. Wolfram Alpha apparently doesn't know about the relativistic rocket equation. Try here:
http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.h...
Generally when we talk about an object's acceleration, we are talking about its acceleration in its own reference frame. By definition, an object's velocity in its own reference frame is always 0, so acceleration does not become more difficult (ignoring the fact that the surrounding matter will be impacting you at higher speeds). From the perspective of an outside observer who is not accelerating, time would appear to slow down on the accelerating object, and the outside observer would record a decreasing acceleration.
2) Even though you don't observe yourself moving at increasing speeds, you do observe the rest of the universe moving at increasing speeds. This means that small particles impacting your ship will hit it with increasing energy.
Jokes aside, there's no fuel source we know of that would allow a ship to accelerate continuously for 50 years at 1g.
After two years: 89.9%
After four years: 97.2%
After eight years: 99.3%
[1] https://www.google.com/search?q=distance+to+alpha+centauri