coramprat sounds about right there.
Are you sure the alt tested good..? How is the battery exactly going dea, you mean like overnite when the car is off..it wont start in the morning..?
If that's the case the battery is not getting charged, OR more likely the battery is FHUGGED. Once a battery goes dead, it's DEAD...PERIOD. You can recharge it, but it is all FHUGGED up inside. A deep cycle battery can be re-lifed.
If it's the stock battery, buy a new one. Battery's are only good for 3, 4 years TOPS...trust me on that. If you can't find the problem get a trusted...TRUSTED mechanic to look her over, not one that's gonna say you need a new alt/batt etc.
BTW not to sound negative, but 01 was a long time ago, alts and batts suck in general and should be changed if you have the spare $$$ laying around, it's easy enough..
PS i never heard anything called a battery saver relay...? It's either something draining the battery, or the battery is gone and wont hold a charge...i seen this ALL the time at autozone, guy thought the battery was good after 5 years...wanted us to charge them, we did, then they got home, put in the battery and it died the next day.