Maximal Independent Subsets of Linear Spaces

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Maximal Independent Subsets of Linear Spaces
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Whats 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)

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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?

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In General?

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

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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 -gt i is in the independent set
• Xi 0 -gt i is not in the independent set
• have the following constraints
• (X0 X1 X2) lt 3
• (X0 X3 X6) lt 3
• (X0 X5 X6) lt 3
• (X2 X3 X4) lt 3
• (X1 X4 X6) lt 3
• (X2 X6 X5) lt 3
• (X1 X3 X5) lt 3
• maximise X0 X1 X2 X3 X4 X5 X6
• use a constraint programming toolkit

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So?

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

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Thats all