JUSTIN BIEBER and SELENA GOMEZ have broken up.  A source says, quote, "Because of their crazy schedules, it was getting harder and harder to maintain a relationship."

--But it might not be that simple.  Other sources say it was Selena who did the dumping around Halloween . . . and it was because of Justin's new "friendship" with a Victoria's Secret model named Barbara Palvin.

--On Thursday night, they went to see "The Lion King" on Broadway with a group of people.  But afterward, Barbara Tweeted, quote, "hey everyone. please calm down.  he is all yours!! :)  please :)."

--And she later followed up with this Tweet, quote, "There is nothing going on with him.  I met him and did a pic with him.  Wouldn't you do the same?  #friendship"

--The night before, Justin performed at the Victoria's Secret Fashion Show. 

--And after the show, Barbara Tweeted, quote, "okay sooo because u guys were askin for it! with ‪@justinbieber ! you were great tonight!"  She added this photo of herself with Justin.

--A source says Justin was playing the field throughout the fashion show . . . quote, "He was asking for phone numbers from the girls . . . He was hitting on [them] all day.”

--In an interview on Saturday, Justin seemed a little confused about how things are going to work out.

--When asked about the breakup, he said, quote, "I don't know what to say.  I don't know what's going on in my life.  To even assess that, it doesn't make sense 'cause I have not made any comment."

--A source has told TMZ that Justin doesn't want the breakup . . . and he's confident he and Selena will get back together.

(--Meanwhile, Justin performed an acoustic version of JUSTIN TIMBERLAKE'S "Cry Me a River" yesterday in Boston.  Here's video.)

(--Afterward, he Tweeted, quote, "Thank you Boston.  Focused on the positive.  #30MillionBeliebers with me.  I'm never leaving you because you are always there for me.")