Algorithms for Drawing Graphs: an Annotated - Brown University
IV Latin-American Algorithms, Graphs and Optimization Symposium - 2007
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Transcript of IV Latin-American Algorithms, Graphs and Optimization Symposium - 2007
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IV Latin-American Algorithms, Graphs and Optimization Symposium - 2007
Puerto Varas - Chile
The Generalized Max-Controlled The Generalized Max-Controlled Set ProblemSet Problem
Carlos A. MartinhonFluminense Fed. University
Ivairton M. Santos - UFMTLuiz S. Ochi – IC/UFF
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ContentsContents
1. Basic definitions1. Basic definitions
2. The Generalized Max-Controlled Set 2. The Generalized Max-Controlled Set ProblemProblem
3.3. a) A 1/2-Approximation procedurea) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristicb) A Based LP Prog. Heuristic
c) A combined heuristicc) A combined heuristic
5. Comp. results and final comments 5. Comp. results and final comments
4. Tabu Search Procedure4. Tabu Search Procedure
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ContentsContents
1. Basic definitions1. Basic definitions
2. The Generalized Max-Controlled Set 2. The Generalized Max-Controlled Set ProblemProblem
3.3. a) A 1/2-Approximation procedurea) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristicb) A Based LP Prog. Heuristic
c) A combined heuristicc) A combined heuristic
5. Comp. results and final comments 5. Comp. results and final comments
4. Tabu Search Procedure4. Tabu Search Procedure
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Basic definitionsBasic definitions Consider G=(V,E) a non-oriented graph and
MV.
Definition: v is controlled by MV |NG[v]M| |NG[v]|/2
ExampleM
v1
v2
v3 v4
v5
v6 v7Cont(G,M)
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Basic definitionsBasic definitions
• ContCont((G,MG,M) ) → set of vertices controlled by M→ set of vertices controlled by M..
• MM defines a defines a monopolymonopoly in in GG ContCont((G,MG,M) = ) = V.V.
0 1 2
3 4 5
M
Given G=(V,E) and MV:
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Basic definitionsBasic definitions Sandwich Graph
0 1 2
3 4 5
G1=(V,E1)
0 1 2
3 4 5
G=(V,E) where E1E E2
0 1 2
3 4 5
G2=(V,E2)
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Basic definitionsBasic definitions
Monopoly Verification Problem – MVP
• Given GGiven G11(V,E(V,E11), G), G22(V,E(V,E22) and M) and MV, V,
G=(V,E) s.t. E1 E E2 and M is monopoly nopoly in in GG ? ?
• Solved in polynomial time (Makino, Yamashita, Solved in polynomial time (Makino, Yamashita, Kameda, Kameda, AlgorithmicaAlgorithmica [2002]). [2002]).
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Basic definitionsBasic definitions
- Max-Controlled Set Problem – MCSP• If the answer to the MVP is If the answer to the MVP is NO,NO, we have the we have the
MCSP!MCSP!
• In the MCSP, we hope to maximize the In the MCSP, we hope to maximize the
number of vertices controlled by M.number of vertices controlled by M.
• The MCSP is NP-hard !! (Makino The MCSP is NP-hard !! (Makino et alet al..
[2002]).[2002]).
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Basic definitionsBasic definitions MCSP
0 1 2
5
4 6
M
Fixed EdgesOptional Edges
Not-controlled vertices
Controlled vertices
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ContentsContents
1. Basic definitions1. Basic definitions
2. The Generalized Max-Controlled Set 2. The Generalized Max-Controlled Set ProblemProblem
3.3. a) A 1/2-Approximation procedurea) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristicb) A Based LP Prog. Heuristic
c) A combined heuristicc) A combined heuristic
5. Comp. results and final comments 5. Comp. results and final comments
4. Tabu Search Procedure4. Tabu Search Procedure
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GMCSPGMCSP
f-controlled vertices
• A vertex A vertex iiVV is is -controlled by -controlled by MMV V iffiff, |, |
NNGG[[ii]]MM|-||-|NNGG[[ii]]UU| | i i , , withwith i i ZZ and and UU==V V \ \ M.M.
Vertices not -controlled by M-controlled vertices by M
0 1 2
3 4 5
M(0) (4) (1)
(3) (-2) (4)
f i fixed gaps (for i V)
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GMCSPGMCSP
We also add positive weights
0 1
4 52 3
M
(0)[2] (0)[3]
(0)[5] (0)[7] (0)[10](0)[1]
Fixed EdgesOptional Edges
Vertices not -controlled-controlled vertices
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GMCSPGMCSP
Generalized Max-Controlled Set Problem
• INPUT:INPUT: Given G Given G11(V,E(V,E11), G), G22(V,E(V,E22) and M) and MV V (with fixed gaps and positive weights).(with fixed gaps and positive weights).
• OBJECTIVE:OBJECTIVE: We want to find a sandwich We want to find a sandwich graph graph G=(V,E), in order to maximize the sum of the weights of all vertices f-controlled by M.
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GMCSPGMCSP Reduction Rules:
We fix alloptional edges
We deleteall optional edges
M U=V\M
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GMCSPGMCSP Reduction Rules
0 1 2
3 4 5
M(0)[1] (0)[1] (0)[1]
(0)[1] (0)[1] (0)[1]
E1D(M,M) E E1D(M,M)D(U,M)
Fixed EdgesOptional Edges
Vertices not -controlled-controlled vertices
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GMCSPGMCSP Reduction Rules
• Consider the following partition of Consider the following partition of VV::
– MMACAC and and UUAC AC vertices always vertices always -controlled -controlled
– MMNCNC and and UUNC _NC _ vertices never vertices never -controlled -controlled
– MMRR and and UUR R vertices vertices -controlled or not.-controlled or not.
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GMCSPGMCSP
Reduction Rules
MAC
MR
MNC
UAC
UR
UNC
M U
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GMCSPGMCSP
Reduction Rules
MAC
MR
MNC
UAC
UR
UNC
M U
optional edges
fixed edges
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PMCCGPMCCG Reduction Rules
0 1 2
3 4 5
M(0)[1] (0)[1] (0)[1]
(0)[1] (0)[1] (0)[1]
MSC={1}
UNC={5}
Fixed EdgesOptional Edges
Vertices not -controlled by M-controlled vertices by M
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ContentsContents
1. Basic definitions1. Basic definitions
2. The Generalized Max-Controlled Set 2. The Generalized Max-Controlled Set ProblemProblem
3.3. a) A 1/2-Approximation procedurea) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristicb) A Based LP Prog. Heuristic
c) A combined heuristicc) A combined heuristic
5. Comp. results and final comments 5. Comp. results and final comments
4. Tabu Search Procedure4. Tabu Search Procedure
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GMCSPGMCSP ½-Approximation algorithm - GMCSP
• Algorithm 1Algorithm 1
1: 1: WW11 Summation of all weights for Summation of all weights for EE==EE11
2: 2: WW22 Summation of all weights for Summation of all weights for EE==EE22
3: 3: zzH1H1 maxmax{{WW11,,WW22}}
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M(0)[5] (0)[1] (0)[3]
(0)[2] (0)[1] (0)[3]
GMCSPGMCSP
½-approximation for the GMCSP
Not -controlled vertices
f-controlled verticesFixed EdgesOptional Edges
0 1 2
3 4 5
W1=9
W2=7
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ContentsContents
1. Basic definitions1. Basic definitions
2. The Generalized Max-Controlled Set 2. The Generalized Max-Controlled Set ProblemProblem
3.3. a) A 1/2-Approximation procedurea) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristicb) A Based LP Prog. Heuristic
c) A combined heuristicc) A combined heuristic
5. Comp. results and final comments 5. Comp. results and final comments
4. Tabu Search Procedure4. Tabu Search Procedure
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GMCSPGMCSP LP formulation
• Consider Consider KK=|=|VV|+|+maxmax{|{|ii| s.t. | s.t. iiVV}}
Vi
ii zpz maxmax
VizK
fxaxa
iMj Uj
iijijijij
,1
Subject to:
1),(,1 Ejixij Vixii ,1
12 \),(},1,0{ EEjixij
Vizi },1,0{
P~
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GMCSP GMCSP
• ConsiderConsider RRMj
iUj
ijijijiji UMifxaxab
,
M(2)
M
(1)
bi=3 bi=3
1),(,1 Ejixij
Vixii ,1
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PMCCGPMCCG Stronger LP Formulation
Vi
ii zpz maxmax
RRii
Mj Ujiijijijij
UMizb
fxaxa
,1
Subject to:
1),(,1 Ejixij
Vixii ,1
12 \),(},1,0{ EEjixij
Vizi },1,0{
ACACi UMiz ,1
NCNCi UMiz ,0
P
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Theorem Theorem : Let and the optimum : Let and the optimum
values of and respectively. Then:values of and respectively. Then:
GMCSPGMCSP
max~z
maxz
P~
P
maxmax~ zz
max~z
maxz
Z*=? Optimum objective value
What about the feasible solutions?
max
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GMCSPGMCSP
Theorem:Theorem: Consider a relaxed solution of Consider a relaxed solution of
with with .. and . and .
If for some (i,j)If for some (i,j)EE22, then there exists , then there exists
another relaxed solution withanother relaxed solution with
and and
),( zx P
2),(],1,0[ Ejixij Vizi ],1,0[
)1,0(ijx
)ˆ,ˆ( zx
2),(},1,0{ˆ Ejixij Vizi ],1,0[ˆ
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PMCCGPMCCG Feasible solution based in the Linear
Relaxation
0 1 2
3 4
M
0,5
0,5
0,5
0,5
0 1 2
3 4
M
10
0
1
12 \),(,5,0 EEjixij 12 \),(},1,0{ˆ EEjixij
Fixed edgesOptional edges
Not-controlled vertices
Controlled vertices
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Integer solution obtained from our stronger Linear Programming formulation.
• Algorithm 2Algorithm 2
– Given a relaxed solution for .Given a relaxed solution for .
– Define as Define as -controlled all vertice -controlled all vertice iiV V with with
, and not , and not -controlled if-controlled if . .
GMCSPGMCSP
),( zx P
1iz 1iz
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Quality of upper and lower bounds
generated by our stronger formulation P
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ContentsContents
1. Basic definitions1. Basic definitions
2. The Generalized Max-Controlled Set 2. The Generalized Max-Controlled Set ProblemProblem
3.3. a) A 1/2-Approximation procedurea) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristicb) A Based LP Prog. Heuristic
c) A combined heuristicc) A combined heuristic
5. Comp. results and final comments 5. Comp. results and final comments
4. Tabu Search Procedure4. Tabu Search Procedure
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MCSPMCSP
• Combined HeuristicCombined Heuristic - CH- CH
• 1) 1) zz11 ½-approximation ½-approximation
• 2) 2) zz22 Based LP Heuristic Based LP Heuristic
• 3) z 3) z max{ max{zz11 , , zz22}}
((Martinhon&Protti, Martinhon&Protti, LNCCLNCC[2002]) [2002])
4,)1(2
1
2
1
nn
n
MCSP Similar combined heuristic with ratio:
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ContentsContents
1. Basic definitions1. Basic definitions
2. The Generalized Max-Controlled Set 2. The Generalized Max-Controlled Set ProblemProblem
3.3. a) A 1/2-Approximation procedurea) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristicb) A Based LP Prog. Heuristic
c) A combined heuristicc) A combined heuristic
5. Comp. results and final comments 5. Comp. results and final comments
4. 4. Tabu Search ProcedureTabu Search Procedure
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ContentsContents
1. Basic definitions1. Basic definitions
2. The Generalized Max-Controlled Set 2. The Generalized Max-Controlled Set ProblemProblem
3.3. a) A 1/2-Approximation procedurea) A 1/2-Approximation procedure
b) A Based LP Prog. Heuristicb) A Based LP Prog. Heuristic
c) A combined heuristicc) A combined heuristic
5. Comp. results and final comments 5. Comp. results and final comments
4. Tabu Search Procedure4. Tabu Search Procedure
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Computational ResultsComputational Results Tabu Search solutions for instances with
50, 75 and 100 vertices.
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THANK YOU !!
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GMCSPGMCSP Reduction Rules
• Rule 3Rule 3: Add to : Add to EE11 all edges of D( all edges of D(MMACACMMNCNC, U, URR).).
• Rule 4Rule 4: Remove from : Remove from EE22 the edges the edges
DD((MMRR,U,UACACUUNCNC).).
• Rule 5Rule 5: Add or remove at random the edges : Add or remove at random the edges
D(D(MMACACMMNCNC, U, UACACUUNCNC).).
MAC
MR
MNC
UAC
UR
UNC
M U
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GMCSPGMCSP
Reduction Rules
• Given two graphs Given two graphs GG11 e e GG22, and 2 subsets , and 2 subsets A,BA,BVV, ,
we define: we define:
DD((A,BA,B)={()={(i,ji,j))EE22\\EE11 | | iiAA, , jjBB}}
• Rule 1Rule 1:: Add to Add to EE11 the edges the edges DD((M,MM,M).).
• Rule 2Rule 2:: Remove from Remove from EE22 the edges the edges DD((U,UU,U).).