Question

In a recent competition organized by “AmDocs” , I came across the following question : (The basic Idea of the question)

You are a given a matrix of fixed size 12×12.
You are given six line segments of length 6,5,5,4,3,2.
The matrix has empty spaces and filled spaces.
You have to return “Yes” Or “No” , whether all the 6 line segments can be fit into the matrix simultaneously or not.
The lines can be placed horizontally or vertically Only.

What algorithm should be used to solve this problem ? Packing ? Knapsack ?

Answer 1

I would map the problem to SAT and use a SAT solver. There is a pretty natural mapping. Define the variables:

x_s_i_j_d = segment s starts at coordinates (i,j) and goes in direction d

(d is “right” or “down”)

First, iterate over all the segments and starting positions, and see which are viable given the starting matrix.
example, M:

000000000000
111111111111
...

if segment 1 is length 2, then L_seg1_0_0_down = false, because it hits a filled space.

Then, write clauses that prohibit two crossing segments. If segment 1 and segment 2 are both length 2, then we add the clause:

(!L_seg1_0_0_right || !L_seg2_1_0_right)

because if segment 1 uses coordinates (0,0), and (1,0), then segment 2 can’t use (1,0) also.

finally, add the condition that each segment must be used at least once:

(L_seg1_0_0_right || L_seg1_0_1_right || ...)

for all the positions that seg1 can go. Then throw your favorite SAT solver at it.