The main thing is to take redistricting out of the hands of partisan legislatures and into the hands of procedurally neutral commissions of some sort. It's a basic democratic principle: the management of the election shouldn't be in the hands of one of the parties competing in it. I think this is much more important than developing an exact mathematical definition of a "fair" map.
I can understand the existence of criticisms of the various commission-drawn maps, but these criticisms only make sense against the background of a much higher expectation of fairness for the commissions. Any of AZ, WA or NJ would be considered at most a very mild gerrymander if they were instituted by a legislature of the party on whom they are supposed to confer advantage.
Better still is to have the commission sort from publicly generated maps. The MN contest drew 500 submissions after the OH contest had 100 entries. The public can draw good maps, and with some guidelines as to the goals from the commission then the effects of any commission bias can be reduced.
Whomever sets the "guidelines," in effect, will draw the lines through the surrogates of his choosing. Any "contest" merely will offer the veneer of democratic participation in what is really an elite-driven process.
Which maps submitted by the public are "good" depends a lot on your point of view. One person might view maps that are compact and rather squarish "good," while another might view maps that meander to group together "communities of interests," whatever that means, as "good."
Many of the problems associated with redistricting are moral. We have elected politicians whom will not pursue the general interest. The solution to many of the problems associated with redistricting is going to have to be an equally moral solution. It's hard to legislate morality, and, it is nearly impossible to mathematically quantify morality.
