Maximal Independent Subsets of Linear Spaces

What’s a linear space?

• Given • a set of points V• a set of lines

• where a line is a k-set of points• each pair of points occurs in exactly one line

• Find• the largest set of independent points IP

• such that IP does not subsume any line (k-set)

An Example

V = {0,1,2,3,4,5,6}

L = {{0,1,2},{0,3,6},{0,4,5},{2,3,4},{1,4,6},{2,6,5},{1,3,5}}

IP = {0,1,3,4}

• L is a design• there are many designs• you are given the design

• IP is an independent set of points• but is it maximal?

In General?

The line can be of size k with property no two lines contain the same pair each pair exists

A Solution? V = {0,1,2,3,4,5,6}

L = {{0,1,2},{0,3,6},{0,5,6},{2,3,4},{1,4,6},{2,6,5},{1,3,5}}

IP = {0,1,3,4}

using the above as an example have variables X0, X1, X2, X3, X4, X5, X6 Xi = 1 -> i is in the independent set Xi = 0 -> i is not in the independent set have the following constraints

(X0 + X1 + X2) < 3 (X0 + X3 + X6) < 3 (X0 + X5 + X6) < 3 (X2 + X3 + X4) < 3 (X1 + X4 + X6) < 3 (X2 + X6 + X5) < 3 (X1 + X3 + X5) < 3

maximise X0 + X1 + X2 + X3 + X4 + X5 + X6 use a constraint programming toolkit

So?

The encoding ignores possibly useful information What variable and value ordering heuristics might we use? Where’s the data Where’s the constraint program (any takers?)? Is it interesting? Is the design more interesting than the independent set?

That’s all