Rook Polynomial Relaxation Labeling Ofir Cohen Shay Horonchik

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

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

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

Rook Polynomial via Relaxation Labeling Compatibility -  Example:

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)

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

Implementation  Input:  Number Of Columns  Number Of Rows  Number Of Cells With Holes

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”

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