How can I turn on and off a switch every 20 minutes?
Oh dear no.  Please use a microcontroller.

The Microcontroller

Look for one that has a timer (I don't think I've ever seen one that doesn't).  Your timer will overflow at a periodic, predictable interval (which you will set up).  Divide 20 minutes by that interval and you will know the number of timer overflows needed to "count" to 20 minutes.  Each time the timer interrupt fires, increment a variable.  When the variable gets to the previously calculated value, switch whatever switch/relay/FET you have for controlling the circuit (see below for that problem).  This solution makes changing the on/off time relatively easy since all you'll be changing is your increment variable max count, rather than having to change the caps &amp; resistors.  You could even have different on/off times if you decide you want that later on, for example: On for 5 mins/off for 15.

The Switch

From what you say, you're looking at 5-15 Amps of current.  The trouble you'll find is that the "inrush current" (which is the sudden surge of current you will get right after the switch is closed).  There are many switches/relays out there that can easily handle 15A of constant current (steady state), but you'll find repeated switching will make them fail because in reality they see a much higher inrush current.  So you'll need to find one that can handle, not just 15A, but whatever inrush current your circuit has.  There are two general categories of switches:


Mechanical - electro-mechanical relays
Solid State - FETs, Triacs, solid-state relays


Of those two, for 15A, I recommend the second category because the mechanical relays will often "arc" when closing to switch on which, over time, produces a buildup on the terminals and they will ether fuse closed (always on) or never be able to completely close (never turn on) if they are switched too often.  Usually you can get a good power-MOSFET or solid-state relay that can properly handle and dissipate the heat associated with the surge of inrush current.  You'll easily find a circuit for how to drive a MOSFET switch from one of the pins on the microcontroller by simply Googling.