Lights Out for Fun and Profit! Parity Domination: Algorithmic and Graph Theoretic Results

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Lights Out for Fun and Profit! Parity Domination: Algorithmic and Graph Theoretic Results. William F. Klostermeyer University of North Florida. Introduction. Green Vertex Pushed. Introduction cont. Of the 2 16 initial configurations of 4 X 4 grid, 2 12 can be changed to all-off

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### Lights Out for Fun and Profit!Parity Domination:Algorithmic and Graph Theoretic Results

William F. Klostermeyer

University of North Florida

Introduction cont.
• Of the 216 initial configurations of 4 X 4 grid, 212 can be changed to all-off
• How many can be changed to all-off in N X N grid?
History
• Lights Out! (~ 1995)
• ACM Programming Contest
• Cellular Automata (1989)
• Parity Domination (1990’s)
Overview
• Complete Solvability
• Fibonacci Polynomials
• Maximization Problems
• Complexity
• Approximation Algorithm
• Fixed Parameter Problems
Parity Domination

1

1

0

1

0

1

p(v) indicated for each v

Parity Domination cont.
• Even Dominating Set:
• Non-empty set of vertices D s.t. each vertex is adjacent to an even number of vertices of D
• Odd Dominating Set:
• Defined accordingly
Parity Domination cont.
• Theorem (Sutner): Every graph has an odd dominating set
• Theorem (folklore): Every initial configuration of G can be turned off iff G has no even dominating set
Even Dominating Sets
• If G has even dominating set, D, closed neighborhood matrix is singular
• Pushing D and empty set have same effect : no change!
• Which graphs have even dominating sets?
Even Dominating Set cont.

0 0 0

0 1 0

1 1 1

0 0 0

Nullspace Matrix

Basics
• Can decide in polynomial time if G has an even dominating set
• use Gaussian elimination
• If G does not have an even dominating set we say G is completely solvable
Basics cont.
• If G has an even dominating set:
• Can decide in polynomial time if a given configuration can be turned off (use linear algebra methods)
Linear Equations

1 1 0 1 0 0 0 0 0 x1 = 1

1 1 1 0 1 0 0 0 0 x2 = 0

0 1 1 0 0 1 0 0 0 x3 = 0

1 0 0 1 1 0 1 0 0 x4 = 0

0 1 0 1 1 1 0 1 0 x5 = 0

0 0 1 0 1 1 0 0 1 x6 = 1

0 0 0 1 0 0 1 1 0 x7 = 0

0 0 0 0 1 0 1 1 1 x8 = 1

0 0 0 0 0 1 0 1 1 x9 = 0

Grids
• 3 X 3 grid completely solvable
• 4 X 4 grid not completely solvable (= has even dominating set)
• Test if Closed Neighborhood Matrix is singular
• O((nm)3) SLOW!
Nullspace Matrices

1 0 0 1 1’s = Even Dominating

1 1 1 1 Set of 4 X 4 Grid

1 1 1 1

1 0 0 1

“Linearize” this matrix to get a 16 X 1 vector in nullspace of closed neighborhood matrix of 4 X 4 grid

Building Nullspace Matrices

0 0 0 0 0 0 0 0 0

1 1 0 1 0 1 0 1 1

0 0 0 1 0 1 0 0 0

1 1 1 0 0 0 1 1 1

0 1 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0

• Thus 4 X 9 grid is not completely solvable.
• Likewise 9 X 9, 4 X 14, 9 X 14, etc.
Nullspace Recurrence

1 0 0 1 1’s = Even Dominating

1 1 1 1 Set of 4 X 4 Grid

1 1 1 1

1 0 0 1

r[I, j]=r[I-1,j]+r[I-1,j-1]+r[I-1,j+1]+r[I-2,j]

mod 2

Recurrence cont.

Theorem: r[I]=fi(B)w

• r[I] : ith row of nullspace matrix
• fi : ith Fibonacci polynomial
• B : Closed Neighborhood Matrix
• w : initial non-zero vector
Fibonacci Polynomials
• Fn(x) is nth Fibonacci polynomial:

f0=0, f1=1, f i=xf i-1(x) + fi-2(x)

f2=x, f3=x2+1, f4=x3

Example

0 0 0

1 0 0 <-- w

1 1 0

1 0 1 = 1 1 0 1 1 0 1 0 0

( 1 1 1 * 1 1 1 + 0 1 1) * 1 0 0 = w

0 1 1 0 1 1 0 1 1

f3=x2+1

Factored Fibonacci Polynomials
• Implemented (randomized) algorithm to factor polynomials over GF(2) in polynomial time
Factored cont.
• f_2: x

(x)^1

• f_3: x^2 +1

(x +1)^2

• f_4: x^3

(x)^3

Fibonacci Polynomials cont.
• f_5: x^4 +x^2 +1

(x^2 +x +1)^2

• f_6: x^5 +x

(x +1)^4

(x)^1

See my web page for thousands more

More on the Recurrence
• Period: number of rows until row of 0’s
• Recurrence is periodic
• Theorem: Maximum period generated by initial vector <1 0 0 0 …>
• Theorem: Length of period is less than 3*2n/2
Periods
• n=5 24, 12, 8, 6, 4, 3, 2
• n=6 9
• n=7 12, 6, 3
• n=8 28, 14, 7, 4, 2
• n=9 30, 15, 10, 5, 3
• n=10 31
• n=12 63
• n=13 18, 9, 3
More Periods

Maximum periods:

• n=39 120
• n=40 1,048,575
• n=41 4680
• n=46 over 8 million
Divisibility Properties
• Theorem: All periods divide the maximum period
• Theorem: If fn+1(x) has only one non-trivial factor, then there is only one period for vectors of length n
Characterization
• Theorem: m x n grid is completely solvable iff

GCD(fn+1(x+1), fm+1(x))=1

over GF(2)

Fast Algorithm
• Can determine if m X n grid is completely solvable in O(n log2 n) time, n>= m
• Obvious method: O((nm)3) time
Square Grids
• Lemma: f2^k+1(x)f2^k-1(x) is equal to square of product of all irred. polynomials with degree dividing k except for x, over GF(2)
• Theorem: 2k x 2k and 2k-2 x 2k-2 grids not completely solvable for all k > 3
Maximization Problems
• Theorem: Can always get at least mn-m/2 off in m X n grid, n >= m
• Theorem: Exist m X n grids for which some initial configurations can get at most mn - (m/log m) off, n >= m
Graphs
• Play Lights Out! in graph
• Closed neighborhood matrix non-singular iff completely solvable iff no even dominating set
• Maximization problems in graphs
Complexity Results
• Theorem: NP-complete to decide if G can be made to have at least k lights out
• Also NP-complete for planar graphs
• Simple approximation algorithm with performance ratio 2
Max-SNP Hard
• Theorem: Exists e > 0 s.t. no approximation algorithm can have performance ratio less than 1+e unless P=NP
• Is there a better approximation algorithm for planar graphs?
Fixed Parameter Problems
• Can decide in polynomial time if a configuration can be made to have n-c off, for constant c
• Gaussian elimination + brute force
Fixed Parameter cont.
• Can decide in polynomial time if all configurations can be made to have n-c off, for a constant c
• Treat all-off state as codeword of binary code
• Test if covering radius of code is at most c

Large grids, 5 by 5 and larger:

Theorem.(Counting argument).

Unsolvable implies not all initial

have at most one light on.

Trees:always at most leaves/2 on.

Conjecture
• Let fn+1 equal square of irred. Polynomial and m be maximum period of n. Then all initial configurations of m X n grid can be made to have at most 2 vertices on.
• Verified for 8 X 6, 30 X 10, 62 X 12, 512 X 18 using Coding Theory algorithm
Publications
• Characterizing Switch-Setting Problems, Lin. and Mult. Alg. 1997
• Maximization Versions of Lights Out …, Cong. Num. 1998
• Fibonacci Polynomials…, Graphs and Combinatorics, to appear
Related Work
• “The Odd Domination Number of a Graph” Y. Caro and W. Klostermeyer, to appear in J. Comb. Math. & Comb. Comput.
• Study size of smallest odd dominating set in graph