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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

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

William F. Klostermeyer

University of North Florida

introduction cont
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
History
  • Lights Out! (~ 1995)
  • Button Madness (PC Game)
  • ACM Programming Contest
  • Cellular Automata (1989)
  • Parity Domination (1990’s)
overview
Overview
  • Complete Solvability
    • Fibonacci Polynomials
  • Maximization Problems
    • Complexity
    • Approximation Algorithm
    • Fixed Parameter Problems
parity domination
Parity Domination

1

1

0

1

0

1

p(v) indicated for each v

parity domination cont
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 cont1
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
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
Even Dominating Set cont.

0 0 0

0 1 0

1 1 1

0 0 0

Nullspace Matrix

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

(x)^1

  • f_3: x^2 +1

(x +1)^2

  • f_4: x^3

(x)^3

fibonacci polynomials cont
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
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
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
More Periods

Maximum periods:

  • n=39 120
  • n=40 1,048,575
  • n=41 4680
  • n=46 over 8 million
divisibility properties
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
Characterization
  • Theorem: m x n grid is completely solvable iff

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

over GF(2)

fast algorithm
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
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
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
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
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
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
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
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
slide39

Large grids, 5 by 5 and larger:

Theorem.(Counting argument).

Unsolvable implies not all initial

configurations can be made to

have at most one light on.

Trees:always at most leaves/2 on.

conjecture
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
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
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