555 Timer PWM generator
R1 is necessary to prevent the 555 attempting to "discharge" the power supply. 

The chosen capacitor charges via R1 + D1 + left-side of VR1.  

The capacitor discharges via right-side of VR1 + D2  + path to ground via pin 7 = discharge.  When pin 7 is low it carries both the capacitor discharge current AND current from V+ via R1. In this case the extra i from R1  = V/R = 9V/1k = 9 mA. 

If R1 was removed it would not change the discharge timing but the capacitor could not charge.

If R1 was shorted it would shorten discharge timing but the IC would short out its own power supply on discharge. The 1K value for R1 is a compromise.

Moving the slider on VR1 from far left to far right changes from a mode of fast charge/slow discharge to slow charge/fast discharge. The charge / discharge pattern is not symmerical as R1 is included in the charge path and not in the discharge path. A more symmetrical effect could be obtained by adding an additional resistor of about 1k in series with D2 - so it altered the discharge path but not the charge path. I say 'about' 1K as the charge discharge operation will never be quite symmetrical.