Greetings, in many of the openings that I play, I end up with a 4:3 kingside pawn majority while my opponent has 2:1 on the queenside. We generally both have at least one rook and a minor piece. What would you suggest is the best way to solve this problem? I have intensively studied the endgame, but to no avail.
This is a very good question as such positions are very common in one's tournament play. As you start getting better and better in chess, the quality of your game will keep going higher. Higher standard generally implies that the times when your opponent handed over the point to you with a silly mistake are long gone and it is time for you to extract that win from no where. Extracting such skillful wins requires a thorough understanding of the position. The position you are talking about in your question is what we refer to as queen side pawn majority positions. Such positions require more of an understanding rather than meticulous calculation.
Queen side majority positions generally work in favor of the side that has the pawn majority on the queen side rather than the king side, since it is much easier to push your extra pawn in the queen side rather than the king side. There are two main factors that are going to influence positions with pawn majorities on opposite sides. The first one would be the control of the central file (most commonly the 'd' file). You might think this is just a simple theory you read in some old chess book a while ago, but in this case it plays a much more important role. Basically, the side that controls the open file will have more chances to control the game and push for a win through-out. The second factor is using your minor piece, in particular your bishops. The long diagonals and the general open nature of the position gives a lot of scope for your bishops.
Let us take a look at some examples to better understand such positions. Our first example is by one of the finest chess players to have ever played the game and a world champion, Alexander Alekhine.
I guess in the end Alekhine made it look pretty simple, going from what looked like a slightly better position to a completely winning position in just no time. Alekhine barely had to break a sweat to win the game.
Our second demonstration is from the man who was known to literally be able to feel his pieces, a world champion with a great natural talent, Jose Raul Capablanca. Let us take a look at how he handled his pawn majority.
If Alekhine made it look simple, Capablanca barely did anything to win this one! Rook penetration and pawn advancement simply did the job for him. The reason for white's quick demise was his completely inactive rook.
Our next example is from another Word champion, Mikhail Botvinnik, who shows his thorough technique to convert this barely equal position.
A very clean execution from Botvinnik, clearly demonstrating the importance of the queen side pawn majority.
The last example we will study today is a demonstration of the bishop's power. This is from one of my own games played against a good friend, GM Davorin Kuljasevich. Once I understood the plan in the middle game it was much easier to play the position.
Again, from nothing to a clearly winning position without doing anything drastic. Of course it still requires a mistake from your opponent to be able to win, but knowing the right plan is the main key to success in such positions.
From the above examples these are the major points that I believe the readers should focus on.
Take control of the 'd' file
The king's activicty is most important in an endgame – Centralize your king
Since these positions tend to be open, bishops have a much better chance against knights
Take control of key squares (against knights in particular) from your opponent's pieces to make them inactive
Hopefully these pointers have helped you understand the queen side pawn majority positions.