IV Latin-American Algorithms,
Graphs and Optimization Symposium - 2007
Puerto Varas - Chile
The Generalized Max-Controlled
Set Problem
Carlos A. Martinhon
Fluminense Fed. University
Ivairton M. Santos - UFMT
Luiz S. Ochi – IC/UFF
Contents
1. Basic definitions
2. The Generalized Max-Controlled Set
Problem
3. a) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristic
c) A combined heuristic
4. Tabu Search Procedure
5. Comp. results and final comments
2
Contents
1. Basic definitions
2. The Generalized Max-Controlled Set
Problem
3. a) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristic
c) A combined heuristic
4. Tabu Search Procedure
5. Comp. results and final comments
3
Basic definitions
 Consider G=(V,E) a non-oriented graph and
MV.
Definition: v is controlled by MV 
|NG[v]M|  |NG[v]|/2
Example
v2
v5
v3
v1
v6
M
v4
v7
Cont(G,M)
4
Basic definitions
Given G=(V,E) and MV:
• Cont(G,M) → set of vertices controlled by M.
• M defines a monopoly in G
Cont(G,M) = V.
M
0
1
2
3
4
5
5
Basic definitions
 Sandwich Graph
0
1
2
0
1
2
3
4
5
3
4
5
G1=(V,E1)
G2=(V,E2)
0
1
2
3
4
5
G=(V,E) where E1E  E2
6
Basic definitions
 Monopoly Verification Problem – MVP
• Given G1(V,E1), G2(V,E2) and MV,
 G=(V,E) s.t. E1  E  E2 and M is monopoly
in G ?
• Solved in polynomial time (Makino, Yamashita,
Kameda, Algorithmica [2002]).
7
Basic definitions
- Max-Controlled Set Problem –
MCSP
• If the answer to the MVP is NO, we have the
MCSP!
• In the MCSP, we hope to maximize the
number of vertices controlled by M.
• The MCSP is NP-hard !! (Makino et
al.[2002]).
8
Basic definitions
 MCSP
M
2
1
0
5
3
4
Fixed Edges
Optional Edges
6
Not-controlled vertices
Controlled vertices
9
Contents
1. Basic definitions
2. The Generalized Max-Controlled Set
Problem
3. a) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristic
c) A combined heuristic
4. Tabu Search Procedure
5. Comp. results and final comments
10
GMCSP
 f-controlled vertices
• A vertex iV is -controlled by MV iff,
|NG[i]M|-|NG[i]U| i , with i Z and U=V \ M.
(0)
0
3
(3)
(1)
(4)
1
4
(-2)
M
f i  fixed gaps (for i  V)
2
5
Vertices not -controlled by M
-controlled vertices by M
(4)
11
GMCSP
 We also add positive weights
(0)[2]
(0)[3]
0
1
2
3
4
(0)[1]
(0)[5]
(0)[7]
Fixed Edges
Optional Edges
M
5
(0)[10]
Vertices not -controlled
-controlled vertices
12
GMCSP
 Generalized Max-Controlled Set Problem
• INPUT: Given G1(V,E1), G2(V,E2) and MV
(with fixed gaps and positive weights).
• OBJECTIVE: We want to find a sandwich
graph G=(V,E), in order to maximize the sum of
the weights of all vertices f-controlled by M.
13
GMCSP
 Reduction Rules:
U=V\M
M
We fix all
optional
edges
We delete
all optional
edges
14
GMCSP
 Reduction Rules
(0)[1]
0
(0)[1]
1
(0)[1] M
2
3
(0)[1]
4
(0)[1]
5
(0)[1]
E1D(M,M)  E  E1D(M,M)D(U,M)
Fixed Edges
Optional Edges
Vertices not -controlled
-controlled vertices
15
GMCSP
 Reduction Rules
• Consider the following partition of V:
– MAC and UAC  vertices always -controlled
– MNC and UNC _  vertices never -controlled
– MR and UR  vertices -controlled or not.
16
GMCSP
 Reduction Rules
MAC
UAC
MR
UR
MNC
UNC
M
U
17
GMCSP
 Reduction Rules
fixed edges
MAC
UAC
MR
UR
MNC
UNC
M
U
optional edges
18
PMCCG
 Reduction Rules
(0)[1]
0
(0)[1]
1
(0)[1] M
2
3
(0)[1]
4
(0)[1]
5
(0)[1]
MSC={1}
UNC={5}
Fixed Edges
Optional Edges
Vertices not -controlled by M
-controlled vertices by M
19
Contents
1. Basic definitions
2. The Generalized Max-Controlled Set
Problem
3. a) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristic
c) A combined heuristic
4. Tabu Search Procedure
5. Comp. results and final comments
20
GMCSP
 ½-Approximation algorithm - GMCSP
• Algorithm 1
1: W1  Summation of all weights for E=E1
2: W2  Summation of all weights for E=E2
3: zH1  max{W1,W2}
21
GMCSP
 ½-approximation for the GMCSP
(0)[5]
0
(0)[1]
1
(0)[3]
M
2
W1=9
W2=7
3
(0)[2]
Fixed Edges
Optional Edges
4
(0)[1]
5
(0)[3]
Not -controlled vertices
f-controlled vertices
22
Contents
1. Basic definitions
2. The Generalized Max-Controlled Set
Problem
3. a) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristic
c) A combined heuristic
4. Tabu Search Procedure
5. Comp. results and final comments
23
GMCSP
 LP formulation P~
• Consider K=|V|+max{|i| s.t. iV}
zmax  max  pi zi
iV
Subject to:
a x  a x
jM
ij ij
jU
ij ij
 fi
K
 1  zi , i V
xij  1, (i, j )  E1
xii  1, i V
xij  {0,1}, (i, j )  E2 \ E1
zi {0,1}, i V
24
GMCSP
• Consider
a x  a x
bi 
jM
ij ij
jU
ij ij
 fi , i  M R U R
xij  1, (i, j )  E1
xii  1,  i V
M
(2)
M
(1)
bi=3
bi=3
25
PMCCG
 Stronger LP Formulation P
zmax  max  pi zi
iV
Subject to:
a x  a x
jM
ij ij
jU
ij ij
 fi
bi
 1  zi , i  M R  U R
zi  1, i  M AC  U AC
zi  0, i  M NC  U NC
xij  1, (i, j )  E1
xii  1, i V
xij  {0,1}, (i, j )  E2 \ E1
zi {0,1}, i V
26
GMCSP
Theorem : Let ~zmax and zmax the optimum
~
values of P and P respectively. Then:
~
zmax  zmax
max
~
zmax
zmax
Z*=?
Optimum objective value
What about the feasible solutions?
27
GMCSP
Theorem: Consider a relaxed solution ( x , z ) of P
with xij  [0,1], (i, j )  E2 and zi  [0,1], i V.
.
If xij  (0,1) for some (i,j)E2, then there exists
another relaxed solution ( xˆ, zˆ) with
xˆij {0,1}, (i, j )  E2 and zˆi  [0,1], i V
28
PMCCG
 Feasible solution based in the Linear
Relaxation
M
0
1
0,5
0,5
2
0
1
0,5
0
1
0,5
3
M
4
xij  0,5, (i, j )  E2 \ E1
Fixed edges
Optional edges
2
1
0
3
4
xˆij  {0,1}, (i, j )  E2 \ E1
Not-controlled vertices
Controlled vertices
29
GMCSP
 Integer solution obtained from our stronger
Linear Programming formulation.
• Algorithm 2
– Given a relaxed solution ( x , z ) for
P
.
– Define as -controlled all vertice iV with
zi  1 , and not -controlled if
zi  1.
30
Quality of upper and lower bounds
generated by our stronger formulation P
31
Contents
1. Basic definitions
2. The Generalized Max-Controlled Set
Problem
3. a) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristic
c) A combined heuristic
4. Tabu Search Procedure
5. Comp. results and final comments
32
MCSP
• Combined Heuristic - CH
• 1) z1  ½-approximation
• 2) z2  Based LP Heuristic
• 3) z  max{z1 , z2}
(Martinhon&Protti, LNCC[2002])
MCSP  Similar combined heuristic with ratio:
1 1 n

, n  4
2 2(n  1)
33
Contents
1. Basic definitions
2. The Generalized Max-Controlled Set
Problem
3. a) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristic
c) A combined heuristic
4. Tabu Search Procedure
5. Comp. results and final comments
34
Contents
1. Basic definitions
2. The Generalized Max-Controlled Set
Problem
3. a) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristic
c) A combined heuristic
4. Tabu Search Procedure
5. Comp. results and final comments
35
Computational Results
 Tabu Search solutions for instances with
50, 75 and 100 vertices.
36
THANK YOU !!
37
GMCSP
 Reduction Rules
• Rule 3: Add to E1 all edges of D(MACMNC, UR).
• Rule 4: Remove from E2 the edges
D(MR,UACUNC).
• Rule 5: Add or remove at random the edges
D(MACMNC, UACUNC).
M
MAC
UAC
MR
UR
MNC
UNC
U
38
GMCSP
 Reduction Rules
• Given two graphs G1 e G2, and 2 subsets A,BV,
we define:
D(A,B)={(i,j)E2\E1 | iA, jB}
• Rule 1: Add to E1 the edges D(M,M).
• Rule 2: Remove from E2 the edges D(U,U).
39