Running an APK without installing it?
No matter if you won't accept the answer, running an APK is not possible.
It's not an executable, it's an archive containing the application and instructions on where all it's resources should be placed so that the app can run.
Android APK File Format
Asking this question is the same as asking how you run an iso image, or how you run a zip, or rar file.
If you really don't want to install it, there's no way to 'run' it.
The only alternative is to install it in a sandbox environment or something like that.
You could try to extract the binary from the APK to the device, and using a shell to run it, but I'm NOT to be held responsible if the results are unwanted.
If you told us why you want to run, but not install it, then I might get you a better answer.