Redundancy Elimination in Proper k-level Hierarchies
and Its Application in a Ramsey Construction
David E. Narv´aez
Department of Computer Sciences
Rochester Institute of Technology
[email protected]
May 15th, 2015
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
1 / 15
Definitions
A directed graph G = (V , E ) is a
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
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Definitions
A directed graph G = (V , E ) is a
k-level hierarchy (Sugiyama et al. , 1981)
if V can be partitioned into k subsets {V0 , V1 , . . . , Vk−1 } such that,
for every edge (a, b) ∈ E ,
if a ∈ Vi and b ∈ Vj then i < j
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
2 / 15
Definitions
A directed graph G = (V , E ) is a
k-level hierarchy (Sugiyama et al. , 1981)
if V can be partitioned into k subsets {V0 , V1 , . . . , Vk−1 } such that,
for every edge (a, b) ∈ E ,
if a ∈ Vi and b ∈ Vj then i < j
Proper k-level hierarchy (Sugiyama et al. , 1981)
if, for every edge (a, b) ∈ E ,
if a ∈ Vi then b ∈ Vi+1
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
2 / 15
Definitions
A directed graph G = (V , E ) is a
k-level hierarchy (Sugiyama et al. , 1981)
if V can be partitioned into k subsets {V0 , V1 , . . . , Vk−1 } such that,
for every edge (a, b) ∈ E ,
if a ∈ Vi and b ∈ Vj then i < j
Proper k-level hierarchy (Sugiyama et al. , 1981)
if, for every edge (a, b) ∈ E ,
if a ∈ Vi then b ∈ Vi+1
Vertices with
in-degree 0 are root nodes.
out-degree 0 are leaf nodes.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
2 / 15
Examples
(a) A single-root example.
Root node is red, leaf nodes
are blue.
David E. Narv´
aez (CS@RIT)
(b) Multiple roots
and leaf nodes in
different levels
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
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Minimization
Is there a subgraph of order m internal nodes where original connectivity between root nodes and leaf nodes is preserved?
Figure: We only need 3 nodes to preserve connectivity.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
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NP-Completeness
Reduction from Hitting-Set (Karp, 1972):
S = {a, b, c, d, e}
C = {{a, b} , {a, b, c} , {b, e} , {d, e}}
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
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NP-Completeness
Reduction from Hitting-Set (Karp, 1972):
S=
C=



a , b , c , d , e
{a, b} , {a, b, c} , {b, e} , {d, e}


David E. Narv´
aez (CS@RIT)





k-level Hierarchies and Ramsey Constructions
May 15th, 2015
6 / 15
NP-Completeness
Reduction from Hitting-Set (Karp, 1972):
S=
C=



a , b , c , d , e
{a, b} , {a, b, c} , {b, e} , {d, e}


David E. Narv´
aez (CS@RIT)





k-level Hierarchies and Ramsey Constructions
May 15th, 2015
6 / 15
NP-Completeness
Reduction from Hitting-Set (Karp, 1972):
S=
C=



a , b , c , d , e
{a, b} , {a, b, c} , {b, e} , {d, e}


David E. Narv´
aez (CS@RIT)





k-level Hierarchies and Ramsey Constructions
May 15th, 2015
6 / 15
MIPP Formulation
x0
x3
xi ∈ {0, 1}
x1
x10
x4
x6
x2
x7
x11
x8
x5
x9
x12
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
7 / 15
MIPP Formulation
x0
x3
xi ∈ {0, 1}
x1
x10
x4
x6
x2
x7
x11
x8
x9
Every path is a product
x5
x3 x10 x2 x9
x12
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
7 / 15
MIPP Formulation
x0
x3
xi ∈ {0, 1}
x1
x10
x4
x6
x2
x7
x11
x8
x9
Every path is a product
x5
And reachability is be expressed as a
sum of products
x3 x10 x2 x9 + x3 x4 x2 x9 ≥ 1
x12
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
7 / 15
MIPP Formulation
For a mixed-integer polynomial programming formulation:
Assign a 0/1 variable xi per node.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
8 / 15
MIPP Formulation
For a mixed-integer polynomial programming formulation:
Assign a 0/1 variable xi per node.
Let Ps,t be the set of paths from s to t.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
8 / 15
MIPP Formulation
For a mixed-integer polynomial programming formulation:
Assign a 0/1 variable xi per node.
Let Ps,t be the set of paths from s to t.
Constraints are:
Minimize
Subject to
P
xi
xi ∈ {0, 1}
∀s |degin (s) = 0
∀tP
|degQ
out (t) = 0 and Ps,t 6= ∅
y ≥1
P∈Ps,t y ∈P
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
8 / 15
Notation
Kn is the complete graph on n vertices.
Cn and Pn are the cycle and paths on n vertices, resp.
A (G , H; n) is a coloring of the edges of Kn that avoids G in the first
color and H in the second color.
Extensible to (G1 , G2 , . . . , Gk ; n).
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
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Ramsey Extension Processes
The goal is to generate a graph with suitable properties by gluing two
graphs from two families. E.g.:
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
10 / 15
Ramsey Extension Processes
The goal is to generate a graph with suitable properties by gluing two
graphs from two families. E.g.:
Generate (K4 , K5 ; 24) graphs by gluing (K3 , K5 ; i) and (K4 , K4 ; j) graphs
(McKay & Radziszowski, 1995).
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
10 / 15
Ramsey Extension Processes
The goal is to generate a graph with suitable properties by gluing two
graphs from two families. E.g.:
Generate (K4 , K5 ; 24) graphs by gluing (K3 , K5 ; i) and (K4 , K4 ; j) graphs
(McKay & Radziszowski, 1995).
Generate (K5 − P3 , K5 ; 24) graphs by gluing (K5 − P3 , K4 ; i) and
(K4 − P3 , K5 ; j) graphs (Calvert et al. , 2012).
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
10 / 15
Ramsey Extension Processes
The goal is to generate a graph with suitable properties by gluing two
graphs from two families. E.g.:
Generate (K4 , K5 ; 24) graphs by gluing (K3 , K5 ; i) and (K4 , K4 ; j) graphs
(McKay & Radziszowski, 1995).
Generate (K5 − P3 , K5 ; 24) graphs by gluing (K5 − P3 , K4 ; i) and
(K4 − P3 , K5 ; j) graphs (Calvert et al. , 2012).
Generate K4 -free graphs whose complement can be colored with two colors
avoiding monochromatic C4 ’s by gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
10 / 15
Ramsey Extension Processes
The goal is to generate a graph with suitable properties by gluing two
graphs from two families. E.g.:
Generate (K4 , K5 ; 24) graphs by gluing (K3 , K5 ; i) and (K4 , K4 ; j) graphs
(McKay & Radziszowski, 1995).
Generate (K5 − P3 , K5 ; 24) graphs by gluing (K5 − P3 , K4 ; i) and
(K4 − P3 , K5 ; j) graphs (Calvert et al. , 2012).
Generate K4 -free graphs whose complement can be colored with two colors
avoiding monochromatic C4 ’s by gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
Gluing is taken to be an incremental, expensive procedure.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
10 / 15
Ramsey Extension Processes
Gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
For a graph X of order 11 and a graph Y of order 7:
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
11 / 15
Ramsey Extension Processes
Gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
For a graph X of order 11 and a graph Y of order 7:
Pick a suitable vicinity for the first vertex of Y .
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
11 / 15
Ramsey Extension Processes
Gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
For a graph X of order 11 and a graph Y of order 7:
Pick a suitable vicinity for the first vertex of Y .
Keep picking vicinities that are compatible with previously selected ones.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
11 / 15
Ramsey Extension Processes
Gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
For a graph X of order 11 and a graph Y of order 7:
Pick a suitable vicinity for the first vertex of Y .
Keep picking vicinities that are compatible with previously selected ones.
Check the graph. Abort current combination if checks fail.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
11 / 15
Ramsey Extension Processes
Gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
For the same graph X of order 11 and another graph Y of order 7:
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
12 / 15
Ramsey Extension Processes
Gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
For the same graph X of order 11 and another graph Y of order 7:
Repeat the process.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
12 / 15
Ramsey Extension Processes
Gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
For the same graph X of order 11 and another graph Y of order 7:
Repeat the process.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
12 / 15
Ramsey Extension Processes
Gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
For the same graph X of order 11 and another graph Y of order 7:
Repeat the process.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
12 / 15
Ramsey Extension Processes
Gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
For the same graph X of order 11 and another graph Y of order 7:
Repeat the process.
We had done some of these checks.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
12 / 15
Ramsey Extension Processes
Gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
Redundancy can be eliminated by:
Calculating the graph of all induced subgraphs in (K4 , K6 ; 7) graphs.
0,1
0,1,2
0,1,2,3
David E. Narv´
aez (CS@RIT)
0,1,3,2 0,1,2,3
0,1,2
0,2,1,3
0,1
0,1,2
0,2,1
0,2,1,3
0,1,3,2
1,2,0
0,1,2
2,0,3,1
k-level Hierarchies and Ramsey Constructions
3,0,1,2
0,1,2,3
May 15th, 2015
13 / 15
Ramsey Extension Processes
Gluing (K3 , K6 ; 11) and (K4 , K6 ; 7) graphs.
Redundancy can be eliminated by:
Calculating the graph of all induced subgraphs in (K4 , K6 ; 7) graphs.
Then minimizing the resulting k-level hierarchy and use it as a map.
0,1
0,1,2
0,1,2,3
David E. Narv´
aez (CS@RIT)
0,1,3,2 0,1,2,3
0,1,2
0,2,1,3
0,1
0,1,2
0,2,1
0,2,1,3
0,1,3,2
1,2,0
0,1,2
2,0,3,1
k-level Hierarchies and Ramsey Constructions
3,0,1,2
0,1,2,3
May 15th, 2015
13 / 15
Future Work
Find more applications for this.
Further explore the MIPP formulation.
Path-based partitioning: Find a set of nodes that partition a k-level
hierarchy into subgraphs that have more or less the same amount of
paths.
David E. Narv´
aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
May 15th, 2015
14 / 15
Reference
Calvert, J., Schuster, M., & Radziszowski, S. 2012.
Computing the Ramsey Number R (K5 − P3 , K5 ).
Journal of Combinatorial Mathematics and Combinatorial Computing,
131–140.
Karp, R. 1972.
Reducibility among Combinatorial Problems.
Springer.
McKay, B., & Radziszowski, S. 1995.
R (4, 5) = 25.
Journal of Graph Theory, 19(3), 309–322.
Sugiyama, K., Tagawa, S., & Toda, M. 1981.
Methods for Visual Understanding of Hierarchical System Structures.
IEEE Transactions on Systems, Man and Cybernetics, 11(2),
109–125.
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aez (CS@RIT)
k-level Hierarchies and Ramsey Constructions
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