Although I know this post is a little old, I have my own function that is a little more optimized than the one given above:
function! g:UltiSnips_Complete()
call UltiSnips#ExpandSnippetOrJump()
if g:ulti_expand_or_jump_res == 0
if pumvisible()
return "\<C-N>"
else
return "\<TAB>"
endif
endif
return ""
endfunction
Of course, if you just keep the settings that Joey Liu provided and then just use this function everything will work just perfectly!
EDIT: Also, I use another function to increase back-stepping functionality between YouCompleteMe and UltiSnips. I'll show you what I mean:
function! g:UltiSnips_Reverse()
call UltiSnips#JumpBackwards()
if g:ulti_jump_backwards_res == 0
return "\<C-P>"
endif
return ""
endfunction
Then just put this in your .vimrc:
au BufEnter * exec "inoremap <silent> " . g:UltiSnipsJumpBackwardTrigger . " <C-R>=g:UltiSnips_Reverse()<cr>"
As well as let g:UltiSnipsJumpBackwardTrigger="<s-tab>"
and your set!