iv latin-american algorithms, graphs and optimization symposium - 2007

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








4

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)

5

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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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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- 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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3

Basic definitionsBasic definitions MCSP

0 1 2

5

4 6

M

Fixed EdgesOptional Edges

Not-controlled vertices

Controlled vertices

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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]


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)


Vertices not -controlled-controlled vertices

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• 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}


Vertices not -controlled by M-controlled vertices by M

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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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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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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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4. 4. Tabu Search ProcedureTabu 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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• 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).).

iv latin-american algorithms, graphs and optimization symposium - 2007

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