Linear Equations 1 1 0 1 0 0 0 0 0 x 1 = 1 1 1 1 0 1 0 0 0 0 x 2 = 0 0 1

Report
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 216 initial configurations of
4 X 4 grid, 212 can be changed to
all-off
• How many can be changed to alloff in N X N grid?
History
•
•
•
•
•
Lights Out! (~ 1995)
Button Madness (PC Game)
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
1
p(v) indicated for each v
0
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)
3 X 3 Grid
Linear Equations
110100000
111010000
011001000
100110100
010111010
001011001
000100110
000010111
000001011
x1 = 1
x2 = 0
x3 = 0
x4 = 0
x5 = 0
x6 = 1
x7 = 0
x8 = 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
1001
1111
1111
1001
1’s = Even Dominating
Set of 4 X 4 Grid
“Linearize” this matrix to get a 16 X 1 vector
in nullspace of closed neighborhood
matrix of 4 X 4 grid
Building Nullspace Matrices
000000000
110101011
000101000
111000111
010000010
000000000
• Thus 4 X 9 grid is not completely solvable.
• Likewise 9 X 9, 4 X 14, 9 X 14, etc.
Nullspace Recurrence
1001
1111
1111
1001
1’s = Even Dominating
Set of 4 X 4 Grid
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=x f i-1(x) + fi-2(x)
f2=x, f3=x2+1, f4=x3
Example
000
1 0 0 <-- w
110
101= 110 110 100
( 1 1 1 * 1 1 1 + 0 1 1) * 1 0 0 = w
011 011 011
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
n=6
n=7
n=8
n=9
n=10
n=12
n=13
24, 12, 8, 6, 4, 3, 2
9
12, 6, 3
28, 14, 7, 4, 2
30, 15, 10, 5, 3
31
63
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
configurations can be made to
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

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