Counting maximal independent sets in subcubic graphs.
Counting maximal independent sets in subcubic graphs. Konstanty Junosza-Szaniawski, Michal Tuczy´ nski Warsaw University of Technology
SOFSEM 2012
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known:
Complexity ∆=2 ∆=3
Independent sets polynomial
Maximal independent sets polynomial
∆=4 ∆=5
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known:
Complexity ∆=2 ∆=3 ∆=4 ∆=5
Independent sets polynomial
Maximal independent sets polynomial
#P-complete Vadhan 1997 #P-complete Vadhan 1997
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known:
Complexity ∆=2 ∆=3 ∆=4 ∆=5
Independent sets polynomial
Maximal independent sets polynomial
#P-complete Vadhan 1997 #P-complete Vadhan 1997
#P-complete Vadhan 1997
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known:
Complexity ∆=2 ∆=3 ∆=4 ∆=5
Independent sets polynomial #P-complete Greenhill 2000 #P-complete Vadhan 1997 #P-complete Vadhan 1997
Maximal independent sets polynomial
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
#P-complete Vadhan 1997
Counting maximal independent sets in subcubic graphs.
What is known:
Complexity ∆=2 ∆=3 ∆=4 ∆=5
Independent sets polynomial #P-complete Greenhill 2000 #P-complete Vadhan 1997 #P-complete Vadhan 1997
Maximal independent sets polynomial
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
#P-complete Greenhill 2000 #P-complete Vadhan 1997
Counting maximal independent sets in subcubic graphs.
What is known:
Complexity ∆=2 ∆=3 ∆=4 ∆=5
Independent sets polynomial #P-complete Greenhill 2000 #P-complete Vadhan 1997 #P-complete Vadhan 1997
Maximal independent sets polynomial ???
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
#P-complete Greenhill 2000 #P-complete Vadhan 1997
Counting maximal independent sets in subcubic graphs.
What is known: Moon, Moser 1965 Number of MIS is at most 3n/3 = 1.44..n v A A v Av
v A A v Av
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
r r r
v A A v Av
Counting maximal independent sets in subcubic graphs.
What is known: Moon, Moser 1965 Number of MIS is at most 3n/3 = 1.44..n v A A v Av
v A A v Av
r r r
v A A v Av
Johnson, Yannakakis 1988 MIS can be generated with polynomial delay
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known: Moon, Moser 1965 Number of MIS is at most 3n/3 = 1.44..n v A A v Av
v A A v Av
r r r
v A A v Av
Johnson, Yannakakis 1988 MIS can be generated with polynomial delay Gaspers, Kratsch, Liedloff 2008, 2012 Maximal independent sets can be counted in time O ∗ (1.3642n )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known: Moon, Moser 1965 Number of MIS is at most 3n/3 = 1.44..n v A A v Av
v A A v Av
r r r
v A A v Av
Johnson, Yannakakis 1988 MIS can be generated with polynomial delay Gaspers, Kratsch, Liedloff 2008, 2012 Maximal independent sets can be counted in time O ∗ (1.3642n ) Maximal independent sets in a subcubic graph can be counted in time O ∗ (1.3532n ) in polynomial space.
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known
Bj¨orklund, Husfeldt 2006 Graph can be colored in time O ∗ ((1 + c)n ) and polynomial space, it there is an algorithm counting independent sets in time O ∗ (c n ) and polynomial space.
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known
Bj¨orklund, Husfeldt 2006 Graph can be colored in time O ∗ ((1 + c)n ) and polynomial space, it there is an algorithm counting independent sets in time O ∗ (c n ) and polynomial space. Counting independent sets - not necessary maximal Dahll¨of, Jonsson, Wahlstr¨ om 2002
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
∆(G ) = 3 O ∗ (1.18..n )
arbitrary ∆(G ) O ∗ (1.25..n )
Counting maximal independent sets in subcubic graphs.
What is known
Bj¨orklund, Husfeldt 2006 Graph can be colored in time O ∗ ((1 + c)n ) and polynomial space, it there is an algorithm counting independent sets in time O ∗ (c n ) and polynomial space. Counting independent sets - not necessary maximal Dahll¨of, Jonsson, Wahlstr¨ om 2002 F¨ urer, Kasiviswanathan 2005
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
∆(G ) = 3 O ∗ (1.18..n ) O ∗ (1.15..n )
arbitrary ∆(G ) O ∗ (1.25..n ) O ∗ (1.24..n )
Counting maximal independent sets in subcubic graphs.
What is known
Bj¨orklund, Husfeldt 2006 Graph can be colored in time O ∗ ((1 + c)n ) and polynomial space, it there is an algorithm counting independent sets in time O ∗ (c n ) and polynomial space. Counting independent sets - not necessary maximal Dahll¨of, Jonsson, Wahlstr¨ om 2002 F¨ urer, Kasiviswanathan 2005 Wahlstr¨om 2008
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
∆(G ) = 3 O ∗ (1.18..n ) O ∗ (1.15..n ) O ∗ (1.15..n )
arbitrary ∆(G ) O ∗ (1.25..n ) O ∗ (1.24..n ) O ∗ (1.23..n )
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v P PPv x v P PPv v v v y v PP @ Pv @ v z@ v P PPv
v ∈S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v P PPv x v P PPv v v v y v PP @ Pv @ v z@ v P PPv
v ∈S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v PPPv x v PP Pv v y v v v PP @ Pv @ v z@ v PP Pv
v ∈S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v P PPv x v P PPv v v v y v PP @ Pv @ v z@ v P PPv
v∈ / S, x ∈ S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v PPPv x v PP Pv v y v v v PP @ Pv @ v z@ v P PPv
v∈ / S, x ∈ S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P v P PPPv x v PP Pv v y v v v PP @ Pv @ v z@ v P PPv
v∈ / S, x ∈ S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v P PPv x v P PPv v v v y v PP @ Pv @ v z@ v P PPv
v, x ∈ /S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v P PPv x v P PPv v v v y v PP @ Pv @ v z@ v P PPv
v, x ∈ /S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Reduction
u v v v @ @ @
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Reduction
u v v v
99K
v v @ @
@ @ @
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
@
Counting maximal independent sets in subcubic graphs.
Reduction
u v v v
99K
v v @ @
@ @ @
@
c1 (v ) := c1 (v ) · c0 (u) c0 (v ) := c0 (v ) · (c1 (u) + c0 (u)) c¯0 (v ) := c1 (u) Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Reduction X Y
c1 (w )
S w :w ∈S
Y
c0 (w )
w :w ∈S, / ¯ )∩S6=∅ N(w
Y
c¯0 (w )
¯ ]∩S=∅ w :N[w
For c1 = 1, c0 = 1, c¯0 = 0 the sum is equal to the number of MIS.
u v v v
99K
v v @ @
@ @ @
@
c1 (v ) := c1 (v ) · c0 (u) c0 (v ) := c0 (v ) · (c1 (u) + c0 (u)) c¯0 (v ) := c1 (u) Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
The algorithm
MISCount(G , c1 , c0 , c¯0 ) 1. Reduction(G , c1 , c0 , c¯0 ) 2. Choose a vertex v 3. Branch on the vertex v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
x v v vy v
v
@
v
z v @
v
v PP v PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v PP v PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v PP v PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v PP v PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
T (n3 ) = T (n3 − 4) + T (n3 − 4) + T (n3 − 2)
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
T (n3 ) = T (n3 − 4) + T (n3 − 4) + T (n3 − 2) T (n3 ) = O ∗ (1.41..n3 )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
T (n3 ) = T (n3 − 4) + T (n3 − 4) + T (n3 − 2) T (n3 ) = O ∗ (1.41..n3 ) 1 T (n) = O ∗ (1.41.. 4 ·n ) =
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
T (n3 ) = T (n3 − 4) + T (n3 − 4) + T (n3 − 2) T (n3 ) = O ∗ (1.41..n3 ) 1 T (n) = O ∗ (1.41.. 4 ·n ) = O ∗ (1.09..n )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
x vw v PP v vy v v v P @
@ z v
v
P v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x vw v PP v vy v v v @
@ z v
v
PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x vw v PP v vy v v v @
@ z v
v
PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x vw v PP v vy v v v @
@ z v
v
PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x vw v x vw v PP PP v vy v v v v vy v v v P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S x vw v x vw v PP PP v vy v v v v vy v v v P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S x vw v x vw v PP PP v vy v v v v vy v v v P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S x vw v x vw v PP PP v vy v v v v vy v v v P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S x vw v x vw v x vw v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
@
z@v
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S v, x ∈ /S w x vw v x x vw v v v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
@
z@v
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S v, x ∈ /S w x vw v x x vw v v v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
@
z@v
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S v, x ∈ /S w x vw v x x vw v v v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
v
P v PP
@
z@v
v
P v PP
T (n3 ) = T (n3 − 4) + T (n3 − 6) + T (n3 − 2)
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S v, x ∈ /S w x vw v x x vw v v v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
v
P v PP
@
z@v
v
P v PP
T (n3 ) = T (n3 − 4) + T (n3 − 6) + T (n3 − 2) T (n3 ) = O ∗ (1.36..n3 )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S v, x ∈ /S w x vw v x x vw v v v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
v
P v PP
@
z@v
v
P v PP
T (n3 ) = T (n3 − 4) + T (n3 − 6) + T (n3 − 2) T (n3 ) = O ∗ (1.36..n3 ) 1 T (n) = O ∗ (1.36.. 3.5 ·n ) =
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S v, x ∈ /S w x vw v x x vw v v v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
v
P v PP
@
z@v
v
P v PP
T (n3 ) = T (n3 − 4) + T (n3 − 6) + T (n3 − 2) T (n3 ) = O ∗ (1.36..n3 ) 1 T (n) = O ∗ (1.36.. 3.5 ·n ) = O ∗ (1.09..n )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
u v P P
x vw v PP v vy v P @
P z @ v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
u v PP v ∈S x vw v PP v vy v @
PP z v @ PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
u v PP v ∈S x vw v PP v vy v @
PP z v @ PP
u v PP x ∈S x vw v PP v vy v @
PP z v @ PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
u v PP v ∈S x vw v PP v vy v @
PP z v @ PP
u v PP x ∈S x vw v PP v vy v @
PP z v @ PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
u v PP v, x ∈ /S x vw v PP v vy v @
PP z v @ PP
Counting maximal independent sets in subcubic graphs.
Complexity
u v PP v ∈S x vw v PP v vy v @
PP z v @ PP
u v PP x ∈S x vw v PP v vy v @
PP z v @ PP
u v PP v, x ∈ /S x vw v PP v vy v @
PP z v @ PP
T (n3 ) = T (n3 − 10) + T (n3 − 10) + T (n3 − 2)
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
u v PP v ∈S x vw v PP v vy v @
PP z v @ PP
u v PP x ∈S x vw v PP v vy v @
PP z v @ PP
u v PP v, x ∈ /S x vw v PP v vy v @
PP z v @ PP
T (n3 ) = T (n3 − 10) + T (n3 − 10) + T (n3 − 2) T (n3 ) = O ∗ (1.21..n3 )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
u v PP v ∈S x vw v PP v vy v @
PP z v @ PP
u v PP x ∈S x vw v PP v vy v @
PP z v @ PP
u v PP v, x ∈ /S x vw v PP v vy v @
PP z v @ PP
T (n3 ) = T (n3 − 10) + T (n3 − 10) + T (n3 − 2) T (n3 ) = O ∗ (1.21..n3 ) T (n) = O ∗ (1.21..n )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Lemma on density
2m n
> 2.25
v @ @
v
v PP P
v
v
@v
v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Lemma on density
2m n
> 2.25
v @ @
v
v PP P
v
v
@v
v
v 2m n
>
16 7
v @
v @
@v
v PP P v PP P v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Lemma on density v 2m n
>
7 3
v @
v @
@v
v PP P v PP P v PP P
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v PP Pv v @
v @ @v
v PP P v
Counting maximal independent sets in subcubic graphs.
Lemma on density v 2m n
2m n
>
>
7 3
28 11
v @
v @
@v
@
@v
v @
v PP P v PP P
v PP P v P PP v PP Pv v v PP Pv
v v PP Pv v @
v @ @v
v PP P v
v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Lemma on density
2m n
>
8 3
v P P
@ v P P
P v P PP
v @ @
P
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Lemma on density
2m n
>
8 3
v P P
@ v P P
P v P PP
v @ @
P
X v XX Pv X XX v v PP @ P v @ XXX @v PP Pv X XX v PP
2m n
> 2.8
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity - Measure and Conquer
Let µ be a measure of a graph, let Gi for i = 1..t be graphs obtained in the i-th branch of recursive call of an algorithm A, let τ0 be the largest root of k X
x −(µ(G )−µ(Gi )) = 1
i=1 µ(G )
then the algorithm A runs in time O ∗ (τ0
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
).
Counting maximal independent sets in subcubic graphs.
Complexity - the measure µ0 (n2 , n3 ) µ1 (n2 , n3 ) µ(n2 , n3 ) = µ2 (n2 , n3 ) ... µ7 (n2 , n3 )
if if if
n = 0 lub 2m n =2 1 ≤ 2 2 < 2m n 4 1 2m 2 4 < n ≤ 2 72
if
2 14 17
SOFSEM 2012
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known:
Complexity ∆=2 ∆=3
Independent sets polynomial
Maximal independent sets polynomial
∆=4 ∆=5
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known:
Complexity ∆=2 ∆=3 ∆=4 ∆=5
Independent sets polynomial
Maximal independent sets polynomial
#P-complete Vadhan 1997 #P-complete Vadhan 1997
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known:
Complexity ∆=2 ∆=3 ∆=4 ∆=5
Independent sets polynomial
Maximal independent sets polynomial
#P-complete Vadhan 1997 #P-complete Vadhan 1997
#P-complete Vadhan 1997
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known:
Complexity ∆=2 ∆=3 ∆=4 ∆=5
Independent sets polynomial #P-complete Greenhill 2000 #P-complete Vadhan 1997 #P-complete Vadhan 1997
Maximal independent sets polynomial
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
#P-complete Vadhan 1997
Counting maximal independent sets in subcubic graphs.
What is known:
Complexity ∆=2 ∆=3 ∆=4 ∆=5
Independent sets polynomial #P-complete Greenhill 2000 #P-complete Vadhan 1997 #P-complete Vadhan 1997
Maximal independent sets polynomial
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
#P-complete Greenhill 2000 #P-complete Vadhan 1997
Counting maximal independent sets in subcubic graphs.
What is known:
Complexity ∆=2 ∆=3 ∆=4 ∆=5
Independent sets polynomial #P-complete Greenhill 2000 #P-complete Vadhan 1997 #P-complete Vadhan 1997
Maximal independent sets polynomial ???
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
#P-complete Greenhill 2000 #P-complete Vadhan 1997
Counting maximal independent sets in subcubic graphs.
What is known: Moon, Moser 1965 Number of MIS is at most 3n/3 = 1.44..n v A A v Av
v A A v Av
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
r r r
v A A v Av
Counting maximal independent sets in subcubic graphs.
What is known: Moon, Moser 1965 Number of MIS is at most 3n/3 = 1.44..n v A A v Av
v A A v Av
r r r
v A A v Av
Johnson, Yannakakis 1988 MIS can be generated with polynomial delay
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known: Moon, Moser 1965 Number of MIS is at most 3n/3 = 1.44..n v A A v Av
v A A v Av
r r r
v A A v Av
Johnson, Yannakakis 1988 MIS can be generated with polynomial delay Gaspers, Kratsch, Liedloff 2008, 2012 Maximal independent sets can be counted in time O ∗ (1.3642n )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known: Moon, Moser 1965 Number of MIS is at most 3n/3 = 1.44..n v A A v Av
v A A v Av
r r r
v A A v Av
Johnson, Yannakakis 1988 MIS can be generated with polynomial delay Gaspers, Kratsch, Liedloff 2008, 2012 Maximal independent sets can be counted in time O ∗ (1.3642n ) Maximal independent sets in a subcubic graph can be counted in time O ∗ (1.3532n ) in polynomial space.
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known
Bj¨orklund, Husfeldt 2006 Graph can be colored in time O ∗ ((1 + c)n ) and polynomial space, it there is an algorithm counting independent sets in time O ∗ (c n ) and polynomial space.
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
What is known
Bj¨orklund, Husfeldt 2006 Graph can be colored in time O ∗ ((1 + c)n ) and polynomial space, it there is an algorithm counting independent sets in time O ∗ (c n ) and polynomial space. Counting independent sets - not necessary maximal Dahll¨of, Jonsson, Wahlstr¨ om 2002
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
∆(G ) = 3 O ∗ (1.18..n )
arbitrary ∆(G ) O ∗ (1.25..n )
Counting maximal independent sets in subcubic graphs.
What is known
Bj¨orklund, Husfeldt 2006 Graph can be colored in time O ∗ ((1 + c)n ) and polynomial space, it there is an algorithm counting independent sets in time O ∗ (c n ) and polynomial space. Counting independent sets - not necessary maximal Dahll¨of, Jonsson, Wahlstr¨ om 2002 F¨ urer, Kasiviswanathan 2005
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
∆(G ) = 3 O ∗ (1.18..n ) O ∗ (1.15..n )
arbitrary ∆(G ) O ∗ (1.25..n ) O ∗ (1.24..n )
Counting maximal independent sets in subcubic graphs.
What is known
Bj¨orklund, Husfeldt 2006 Graph can be colored in time O ∗ ((1 + c)n ) and polynomial space, it there is an algorithm counting independent sets in time O ∗ (c n ) and polynomial space. Counting independent sets - not necessary maximal Dahll¨of, Jonsson, Wahlstr¨ om 2002 F¨ urer, Kasiviswanathan 2005 Wahlstr¨om 2008
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
∆(G ) = 3 O ∗ (1.18..n ) O ∗ (1.15..n ) O ∗ (1.15..n )
arbitrary ∆(G ) O ∗ (1.25..n ) O ∗ (1.24..n ) O ∗ (1.23..n )
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v P PPv x v P PPv v v v y v PP @ Pv @ v z@ v P PPv
v ∈S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v P PPv x v P PPv v v v y v PP @ Pv @ v z@ v P PPv
v ∈S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v PPPv x v PP Pv v y v v v PP @ Pv @ v z@ v PP Pv
v ∈S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v P PPv x v P PPv v v v y v PP @ Pv @ v z@ v P PPv
v∈ / S, x ∈ S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v PPPv x v PP Pv v y v v v PP @ Pv @ v z@ v P PPv
v∈ / S, x ∈ S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P v P PPPv x v PP Pv v y v v v PP @ Pv @ v z@ v P PPv
v∈ / S, x ∈ S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v P PPv x v P PPv v v v y v PP @ Pv @ v z@ v P PPv
v, x ∈ /S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Branching
v P P P v P PPv x v P PPv v v v y v PP @ Pv @ v z@ v P PPv
v, x ∈ /S
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Reduction
u v v v @ @ @
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Reduction
u v v v
99K
v v @ @
@ @ @
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
@
Counting maximal independent sets in subcubic graphs.
Reduction
u v v v
99K
v v @ @
@ @ @
@
c1 (v ) := c1 (v ) · c0 (u) c0 (v ) := c0 (v ) · (c1 (u) + c0 (u)) c¯0 (v ) := c1 (u) Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Reduction X Y
c1 (w )
S w :w ∈S
Y
c0 (w )
w :w ∈S, / ¯ )∩S6=∅ N(w
Y
c¯0 (w )
¯ ]∩S=∅ w :N[w
For c1 = 1, c0 = 1, c¯0 = 0 the sum is equal to the number of MIS.
u v v v
99K
v v @ @
@ @ @
@
c1 (v ) := c1 (v ) · c0 (u) c0 (v ) := c0 (v ) · (c1 (u) + c0 (u)) c¯0 (v ) := c1 (u) Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
The algorithm
MISCount(G , c1 , c0 , c¯0 ) 1. Reduction(G , c1 , c0 , c¯0 ) 2. Choose a vertex v 3. Branch on the vertex v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
x v v vy v
v
@
v
z v @
v
v PP v PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v PP v PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v PP v PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v PP v PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
T (n3 ) = T (n3 − 4) + T (n3 − 4) + T (n3 − 2)
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
T (n3 ) = T (n3 − 4) + T (n3 − 4) + T (n3 − 2) T (n3 ) = O ∗ (1.41..n3 )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
T (n3 ) = T (n3 − 4) + T (n3 − 4) + T (n3 − 2) T (n3 ) = O ∗ (1.41..n3 ) 1 T (n) = O ∗ (1.41.. 4 ·n ) =
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x v v vy v @
z v @
v v v
v∈ / S, x ∈ S
v x v PP y v PP v v v @ v z@v PP
v v v
v, x ∈ /S
x v v PP y v PP v v v @ z@v v PP
v v v
v PP v PP v PP
T (n3 ) = T (n3 − 4) + T (n3 − 4) + T (n3 − 2) T (n3 ) = O ∗ (1.41..n3 ) 1 T (n) = O ∗ (1.41.. 4 ·n ) = O ∗ (1.09..n )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
x vw v PP v vy v v v P @
@ z v
v
P v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x vw v PP v vy v v v @
@ z v
v
PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x vw v PP v vy v v v @
@ z v
v
PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x vw v PP v vy v v v @
@ z v
v
PP v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S x vw v x vw v PP PP v vy v v v v vy v v v P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S x vw v x vw v PP PP v vy v v v v vy v v v P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S x vw v x vw v PP PP v vy v v v v vy v v v P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S x vw v x vw v PP PP v vy v v v v vy v v v P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S x vw v x vw v x vw v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
@
z@v
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S v, x ∈ /S w x vw v x x vw v v v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
@
z@v
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S v, x ∈ /S w x vw v x x vw v v v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v
P v PP
@
z@v
v
P v PP
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S v, x ∈ /S w x vw v x x vw v v v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
v
P v PP
@
z@v
v
P v PP
T (n3 ) = T (n3 − 4) + T (n3 − 6) + T (n3 − 2)
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S v, x ∈ /S w x vw v x x vw v v v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
v
P v PP
@
z@v
v
P v PP
T (n3 ) = T (n3 − 4) + T (n3 − 6) + T (n3 − 2) T (n3 ) = O ∗ (1.36..n3 )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S v, x ∈ /S w x vw v x x vw v v v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
v
P v PP
@
z@v
v
P v PP
T (n3 ) = T (n3 − 4) + T (n3 − 6) + T (n3 − 2) T (n3 ) = O ∗ (1.36..n3 ) 1 T (n) = O ∗ (1.36.. 3.5 ·n ) =
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
v ∈S v∈ / S, x ∈ S v, x ∈ /S w x vw v x x vw v v v PP PP PP v vy v v v v vy v v v v vy v v v P P P @
@ z v
v
P v PP
@
z@v
v
P v PP
@
z@v
v
P v PP
T (n3 ) = T (n3 − 4) + T (n3 − 6) + T (n3 − 2) T (n3 ) = O ∗ (1.36..n3 ) 1 T (n) = O ∗ (1.36.. 3.5 ·n ) = O ∗ (1.09..n )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
u v P P
x vw v PP v vy v P @
P z @ v PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
u v PP v ∈S x vw v PP v vy v @
PP z v @ PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
u v PP v ∈S x vw v PP v vy v @
PP z v @ PP
u v PP x ∈S x vw v PP v vy v @
PP z v @ PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
u v PP v ∈S x vw v PP v vy v @
PP z v @ PP
u v PP x ∈S x vw v PP v vy v @
PP z v @ PP
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
u v PP v, x ∈ /S x vw v PP v vy v @
PP z v @ PP
Counting maximal independent sets in subcubic graphs.
Complexity
u v PP v ∈S x vw v PP v vy v @
PP z v @ PP
u v PP x ∈S x vw v PP v vy v @
PP z v @ PP
u v PP v, x ∈ /S x vw v PP v vy v @
PP z v @ PP
T (n3 ) = T (n3 − 10) + T (n3 − 10) + T (n3 − 2)
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
u v PP v ∈S x vw v PP v vy v @
PP z v @ PP
u v PP x ∈S x vw v PP v vy v @
PP z v @ PP
u v PP v, x ∈ /S x vw v PP v vy v @
PP z v @ PP
T (n3 ) = T (n3 − 10) + T (n3 − 10) + T (n3 − 2) T (n3 ) = O ∗ (1.21..n3 )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity
u v PP v ∈S x vw v PP v vy v @
PP z v @ PP
u v PP x ∈S x vw v PP v vy v @
PP z v @ PP
u v PP v, x ∈ /S x vw v PP v vy v @
PP z v @ PP
T (n3 ) = T (n3 − 10) + T (n3 − 10) + T (n3 − 2) T (n3 ) = O ∗ (1.21..n3 ) T (n) = O ∗ (1.21..n )
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Lemma on density
2m n
> 2.25
v @ @
v
v PP P
v
v
@v
v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Lemma on density
2m n
> 2.25
v @ @
v
v PP P
v
v
@v
v
v 2m n
>
16 7
v @
v @
@v
v PP P v PP P v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Lemma on density v 2m n
>
7 3
v @
v @
@v
v PP P v PP P v PP P
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
v v PP Pv v @
v @ @v
v PP P v
Counting maximal independent sets in subcubic graphs.
Lemma on density v 2m n
2m n
>
>
7 3
28 11
v @
v @
@v
@
@v
v @
v PP P v PP P
v PP P v P PP v PP Pv v v PP Pv
v v PP Pv v @
v @ @v
v PP P v
v
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Lemma on density
2m n
>
8 3
v P P
@ v P P
P v P PP
v @ @
P
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Lemma on density
2m n
>
8 3
v P P
@ v P P
P v P PP
v @ @
P
X v XX Pv X XX v v PP @ P v @ XXX @v PP Pv X XX v PP
2m n
> 2.8
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
Counting maximal independent sets in subcubic graphs.
Complexity - Measure and Conquer
Let µ be a measure of a graph, let Gi for i = 1..t be graphs obtained in the i-th branch of recursive call of an algorithm A, let τ0 be the largest root of k X
x −(µ(G )−µ(Gi )) = 1
i=1 µ(G )
then the algorithm A runs in time O ∗ (τ0
Konstanty Junosza-Szaniawski, Michal Tuczy´ nski
).
Counting maximal independent sets in subcubic graphs.
Complexity - the measure µ0 (n2 , n3 ) µ1 (n2 , n3 ) µ(n2 , n3 ) = µ2 (n2 , n3 ) ... µ7 (n2 , n3 )
if if if
n = 0 lub 2m n =2 1 ≤ 2 2 < 2m n 4 1 2m 2 4 < n ≤ 2 72
if
2 14 17
