CANNES -- Lionsgate Entertainment said it will release the third film in its successful "Saw" series Oct. 31 with a substantial publicity campaign, including the release of a special director's cut DVD package earlier in the month. "Saw III" began shooting May 8 in Toronto and should finish by the end of June or early July. Although the studio would not confirm budget figures, they are rumored to be more than $10 million -- more than the first two films combined.