Rook Polynomial Relaxation Labeling

1. Rook PolynomialRelaxation Labeling • Ofir Cohen • Shay Horonchik

2. Problem Domain • Rooks can only move horizontally or vertically. • Place n Rooks on a n*n chess board with holes, where no piece can challenge other rooks. • This is an NP Complete problem

3. Problem Domain (cont.) • Rook Polynomial can be reduced to: • Resource distribution under constraints • Known Solutions • Algorithms using back tracking • Include / exclude mechanism

4. Rook Polynomial via Relaxation Labeling • Set of Objects: • We declared each cell (except holes) as an object. • Set of Labels: • We declared two labels: {Empty, Rook} • Initial Confidence: • Rook => 1 / Maximum between empty cells in row and clumn • Empty => 1 - Empty

5. Rook Polynomial via Relaxation Labeling • Compatibility - • Example:

6. Rook Polynomial via Relaxation Labeling • Results: • Very long running time • it doesn’t converge to the correct solution • The algorithm doesn’t try to achieve maximum rook number on board • Successful runs. (only on small boards)

7. Rook Polynomial via Relaxation Labeling (phase b) • We perform the following changes: • Initial confidence • Randomize rooks on several cells on the board • Support function • Zeroing cells where found rooks in both row and column • Increasing cells value where found an empty row/column

8. Implementation • Input: • Number Of Columns • Number Of Rows • Number Of Cells With Holes

9. Problems And Conclusion • Relaxation algorithm purpose don’t match the problem specification . • Relaxation labeling purpose is to match objects and labels • The rook polynomial problem purpose is to find maximal “Rook labels”

10. Any Questions ?

11. Thank You