Futoshiki Hard

Futoshiki (from Japanese, literally "not equal"; also known as "Hutoshiki", "Unequal") is a logic puzzle. The puzzle is played on a square grid, such as 9 x 9. The objective is to place the numbers 1 to 9 (or whatever the dimensions are) in each row, ensuring that each column also only contains the digits 1 to 9. Some digits may be given at the start. In addition, inequality constraints are also initially specifed between some of the squares, such that one must be higher or lower than its neighbour. These constraints must be honoured as the grid is filled out.