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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. 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

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  1. Lights Out for Fun and Profit!Parity Domination:Algorithmic and Graph Theoretic Results William F. Klostermeyer University of North Florida

  2. Introduction

  3. Green Vertex Pushed

  4. 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?

  5. History • Lights Out! (~ 1995) • Button Madness (PC Game) • ACM Programming Contest • Cellular Automata (1989) • Parity Domination (1990’s)

  6. Overview • Complete Solvability • Fibonacci Polynomials • Maximization Problems • Complexity • Approximation Algorithm • Fixed Parameter Problems

  7. Parity Domination 1 1 0 1 0 1 p(v) indicated for each v

  8. 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

  9. 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

  10. 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?

  11. Even Dominating Set cont. 0 0 0 0 1 0 1 1 1 0 0 0 Nullspace Matrix

  12. 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

  13. 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)

  14. 3 X 3 Grid

  15. 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

  16. 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!

  17. 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

  18. 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.

  19. 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

  20. 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

  21. 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

  22. 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

  23. Factored Fibonacci Polynomials • Implemented (randomized) algorithm to factor polynomials over GF(2) in polynomial time

  24. Factored cont. • f_2: x (x)^1 • f_3: x^2 +1 (x +1)^2 • f_4: x^3 (x)^3

  25. 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

  26. 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

  27. 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

  28. More Periods Maximum periods: • n=39 120 • n=40 1,048,575 • n=41 4680 • n=46 over 8 million

  29. 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

  30. Characterization • Theorem: m x n grid is completely solvable iff GCD(fn+1(x+1), fm+1(x))=1 over GF(2)

  31. 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

  32. 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

  33. 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

  34. Graphs • Play Lights Out! in graph • Closed neighborhood matrix non-singular iff completely solvable iff no even dominating set • Maximization problems in graphs

  35. 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

  36. 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?

  37. 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

  38. 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

  39. 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.

  40. 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

  41. 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

  42. 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

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